Prediction of Residual Liquid Distribution of Austenitic Stainless Steel during Laser Beam Welding Using Multi-Phase Field Modeling

Abstract

The aim of this research was to predict the residual liquid distribution during laser beam welding at different welding speeds using Multi-phase field modeling to investigate the solidification cracking phenomenon. The calculated secondary dendrite arm spacing and primary dendrite tip radius were compared with experimental values and Kurz–Giovanola–Trivedi modeling. The effect of calculation parameters, such as interfacial mobility and anisotropy of interfacial mobility, on lengths of the residual liquid distribution composed of residual liquid connecting with the molten pool (L_P) and film-dot (L_FD) regions was evaluated quantitatively. The length of each region increased with increasing interfacial mobility. The lengths of L_P and L_FD increased and decreased, respectively, with increasing anisotropy of interfacial mobility and finally both remained constant. An adjustment of the calculation parameters using experimental results yielded the lengths of the residual liquid distribution that were nearly the same as that of the fracture surface of a solidification crack. The residual liquid distribution could be predicted from verification with experimental results and calculated parameter optimization at a high cooling rate.

1. Introduction

The susceptibility of type-310S stainless steel to solidification cracking during laser beam welding (LBW) has already been evaluated by the authors quantitatively using a brittleness temperature range (BTR) measured by a two-dimensional (2D) temperature distribution and laser Trans-Varestraint test.¹⁾ However, the solidification cracking mechanism during LBW is still under discussion. In general, solidification cracks occur along solidification grain boundaries,^2,3) and solute redistribution can yield a high concentration of solute and impurity elements along this boundary, which results in the formation of a low-melting residual liquid film along the boundary and induces solidification cracking.³⁾ BTR and its ductility are the two most important indices that are used to evaluate solidification cracking susceptibility, whereas the morphology and distribution of residual liquid in the BTR mainly influence these values at the terminal of solidification. For instance, a residual liquid film lowers the ductility, which increases the solidification cracking susceptibility as a result of the formation of a continuous liquid interface in the BTR.²⁾ Thus, it is necessary to obtain the morphology and distribution of the residual liquid along the solidification grain boundary to understand the solidification cracking mechanism. However, because of the high cooling rate, it is difficult to measure experimentally these distributions and morphologies at the terminal of solidification during LBW. Therefore, simulation can be used to predict the morphology and distribution of the residual liquid.

Several simulation methods exist for the solidification microstructure, such as the deterministic,⁴⁾ probabilistic^5,6,7,8) and phase field methods.^{9,10,11,12,13)} The deterministic method does not consider random phenomena and ignores the crystallographic morphology during dendrite growth.⁴⁾ Thus, it has been difficult to simulate the dendrite morphology precisely. The probabilistic method can be divided into two types, namely, the Monte Carlo and Cellular Automaton.⁷⁾ The Monte Carlo method lacks physical basics, such as nucleation and growth of crystalline,⁶⁾ which results in difficulties in analyzing physical phenomena quantitatively.⁷⁾ Moreover, the simulation result may differ slightly from experimental data using the Cellular Automaton method because few macro factors are considered. The Multi-phase field method (MPFM) is based on the Ginsberg–Landau theory and gives expression to the influence of diffusion, ordering potential, and thermodynamic driving force in terms of differential equations that can simulate microstructure evaluation in time and space by the coupling of phase, temperature, solute, and flow fields.^9,10) Thus, the MPFM is one of the most useful methods to simulate sequences of dendrite growth and residual liquid morphology and distribution during solidification.

To date, the MPFM has been carried out at low cooling rate, such as in casting^14,15) and gas tungsten arc welding.^16,17) In addition, little research exists to verify calculated result with that of the experiment quantitatively, which makes it difficult to fit suitable calculation parameters to predict real solidification phenomenon. Thus, the aim of this work is to use the MPFM to simulate the residual liquid distribution at a high cooling rate, as occurs in LBW. The fracture surface of the solidification crack exhibits a dendritic appearance that corresponds to the residual liquid distribution.³⁾ Therefore, the calculated results will be verified with those of the fracture surface of the solidification crack.

In this study, the MPFM was used to predict the distribution of the residual liquid region and the geometry of type-310S austenitic stainless steel at a high cooling rate during LBW by verification with the experiment and adjusted calculation parameters, where the interfacial mobility corresponds with a kinetic coefficient in the software, and an anisotropy of interfacial mobility. First, the calculated secondary dendrite arm spacing and primary dendrite tip radius were compared with the experimental results and the Kurz–Giovanola–Trivedi (KGT) model, which is a dendritic growth model that was used to verify the calculations.^18,19) Then, the effect of calculation parameters on residual liquid distribution was investigated. Next, the residual liquid distributions were simulated at different cooling rates by verification with that of the fracture surface of the solidification crack and by optimizing calculation parameters to understand the solidification cracking mechanism. Finally, the effect of cooling rate on the calculation parameters was investigated to reveal suitable parameters to predict the real solidification phenomenon at various cooling rates.

2. Calculation Method

2.1. Multi-phase Field Modeling

Phase field method based on Ginsberg-Landau theory,^20,21) a general model of the free energy can be given as an integral of the density functional over the domain Ω by²²⁾   

$$F\left(\left\{{\varphi }_{\alpha }\right\}\right)={\int }_{\mathrm{\Omega }}\left[f^{intf}\left(\left\{{\varphi }_{\alpha }\right\}\right)+f^{chem}\left(\left\{{\varphi }_{\alpha }\right\}\right)\right]$$(1)

where F is the total free energy of the system, ϕ is an ordering variable setting from 0 (liquid phase) to 1 (solid phase), {ϕ_α} is a function of multiple phase fields and the bracket { } denotes all phases α and not an individual α, f^intf is the interfacial free energy density deduced in the literature²¹⁾ and f^chem is the Helmholtz free energy density explained in the following part.

According to a detailed derivation from the literature,²⁰⁾ the governing equation for phase field of α phase can be expressed as^21,22)   

$$\begin{array}{l}\frac{\partial {\varphi }_{\alpha }}{\partial t}={\sum }_{\beta =1}^v{M}_{\alpha \beta }\{{\sigma }_{\alpha \beta }\left[\left({\varphi }_{\beta }{\nabla }^2{\varphi }_{\alpha }-{\varphi }_{\alpha }{\nabla }^2{\varphi }_{\beta }\right)+\frac{{\pi }^2}{2{\eta }^2}\left({\varphi }_{\alpha }-{\varphi }_{\beta }\right)\right]\\ \hspace{1em}\hspace{1em}+\frac{\pi }{\eta }\sqrt{{\varphi }_{\alpha }{\varphi }_{\beta }}\cdot \mathrm{\Delta }{G}_{\alpha \beta }\}\end{array}$$(2)

where ϕ_α and ϕ_β are the phase fields of α and β phases, v is the number of phase, M_αβ is the interfacial mobility, σ_αβ is the interfacial energy, η is the interface thickness and ΔG_αβ is the thermodynamic driving force which can be from a parallel tangent construction²²⁾   

$$\begin{array}{l}\mathrm{\Delta }{G}_{\alpha \beta }=-\left(\frac{\partial }{\partial {\varphi }_{\alpha }}-\frac{\partial }{\partial {\varphi }_{\beta }}\right)f^{chem}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}=\frac{1}{v^m}\left[g_{\beta }\left({\overrightarrow{c}}_{\beta }\right)-g_{\alpha }\left({\overrightarrow{c}}_{\alpha }\right)-\overrightarrow{\tilde{\mu }}\left({\overrightarrow{c}}_{\beta }-{\overrightarrow{c}}_{\alpha }\right)\right]\end{array}$$(3)

  

$$f^{chem}=\frac{g}{v^m}$$(4)

  

$$g\left({\varphi }_{\alpha },{\overrightarrow{c}}_{\alpha }\right)={\sum }_{\alpha =1}^v{\varphi }_{\alpha }g\left({\overrightarrow{c}}_{\alpha }\right)$$(5)

where v^m is the mean molar volume, g_α and g_β are the Gibbs free energy densities of α and β phases, ${\overrightarrow{c}}_{\alpha }=(c_{\alpha }^1,\dots ,c_{\alpha }^n)$ and ${\overrightarrow{c}}_{\beta }$ are the phase compositions, and $\overrightarrow{\tilde{\mu }}$ is the mixture diffusion potential. If the change of the volume is neglected and it is assumed that the molar volumes of all phases are equal, the Helmholtz free energy density can be replaced by the Gibbs free energy density, as shown in Eq. (4).

Then, the diffusion equation can be given by^21,22)   

$$\frac{\partial \overrightarrow{c}}{\partial t}=\nabla \left({\sum }_{\alpha =1}^v{\varphi }_{\alpha }{D}_{\alpha }\nabla {\overrightarrow{c}}_{\alpha }\right)$$(6)

  

$$D_{\alpha }=v^m{M}_{\alpha }^{ch}{T}_{\alpha }$$(7)

  

$$T_{\alpha }^{ij}=\frac{\partial {\tilde{\mu }}_{\alpha }^i}{\partial c_{\alpha }^j}=\frac{{\partial }^2{g}_{\alpha }}{\partial c_{\alpha }^i\partial c_{\alpha }^j}$$(8)

where $\overrightarrow{c}={\sum }_{\alpha =1}^v{\varphi }_{\alpha }{\overrightarrow{c}}_{\alpha }$ is mixture composition, D_α is the diffusion matrix in phase α, $M_{\alpha }^{ch}$ is the chemical mobility matrix which is the function of the atomic mobility and the local phase composition, T_α is the matrix comprised the derivatives of the diffusion potentials in phase α, i and j are the components, ${\tilde{\mu }}_{\alpha }$ is the phase diffusion potential.

As is known, the phase field and diffusion equations need to be solved by quasi-equilibrium data which could be derived from thermodynamic calculations using database.²²⁾ However, the quasi-equilibrium condition should be solved for each location and each interface cell in every numerical time step.²²⁾ Thus, a multi-binary extrapolation is considered in order to improve calculation efficiency^21,22)   

$$c_{\alpha }^i=\frac{c^i-{\sum }_{\rho =1}^v{\varphi }_{\rho }\left[c_{\rho }^{i*}-K_{\rho \alpha }^{ij}{c}_{\alpha }^{i*}+\left(\frac{\partial c_{\rho }^i}{\partial T}\right)\mathrm{\Delta }T\right]}{{\sum }_{\rho =1}^v{\varphi }_{\rho }{K}_{\rho \alpha }^{ij}}$$(9)

where cⁱ is the composition of component i, ρ is an arbitrary reference phase, $c_{\rho }^{i*}$ is the composition of an arbitrary reference phase in quasi-equilibrium condition, $K_{\rho \alpha }^{ij}=\left(\frac{\partial c_{\rho }^i}{\partial c_{\alpha }^j}\right)=T_{\rho }^{-1}{T}_{\alpha }$ is the partition coefficient, ΔT=T−T^* and T^* is the quasi-equilibrium temperature.

MICRESS, which is a commercial software based on phase field modeling for single-, multi-phase, and multi-component systems, was used in this study. A thermodynamic database, Fe (TCFE 7), and a mobility database, MOBFE 2, were applied by direct coupling to the multi-phase field calculation. A multi-obstacle potential was used in the phase field modeling because it takes into account all of the multi-phase or multi-grain interactions and can correct for dihedral angle.²⁰⁾

When the calculation model during solidification of the LBW was considered, several physical and calculation parameters, such as interfacial energy, anisotropy of the interfacial energy, interfacial mobility, and anisotropy of the interfacial mobility were required. Thus, the anisotropy model should also be considered during calculation.

Normally, the anisotropy function of interfacial energy is given by^17,21,23)   

$$\sigma \left(\phi \right)={\sigma }_{i-j}\left[1+{\epsilon }_{4}cos\left(4\phi \right)\right]$$(10)

where i and j represent different phases, ε₄ is the anisotropy coefficient. When the value of ε₄ is more than 0.056, the crystal will change into facet.^17,21) Therefore, in the case of the dendrite growth, ε₄ should set less than 0.056. While, in MICRESS software, the anisotropy of interfacial energy is replaced by the anisotropy of interfacial stiffness because the latter considers a direct matching to the modified Gibbs-Thomson equation for anisotropic interfaces.²⁴⁾ Therefore, for this reason, the anisotropy of interfacial stiffness equals 15×ε₄ in the research.²⁴⁾ In addition, the anisotropy function and relationship are also appropriate for that of the anisotropy of interfacial mobility.

In addition, material parameters, such as the dendrite arm spacing, cooling rate, and temperature gradient during LBW were also required and were measured directly by experiment at first.

2.2. Material Used

Type-310S stainless steel, which had been solidified in γ single mode during LBW, was used. Its chemical composition is shown in Table 1.

Table 1. Chemical composition of Type 310S stainless steel (mass%).

C	Si	Mn	P	S	Ni	Cr	Co	Fe
0.04	0.43	0.96	0.019	0.001	20.13	25.19	0.09	Bal.

2.3. Measurement of Primary and Secondary Dendrite Arm Spacing

To set parameters for the calculations and to verify the calculated results of the MPFM, the primary and secondary dendrite arm spacing needs to be measured experimentally. Figure 1 shows an example of a liquid Sn quenched microstructure at the rear center of the molten pool during LBW at welding speed of 0.2 m/min. In Fig. 1, the primary and secondary dendrite arm spacing can be measured directly from this liquid Sn quenched microstructure; however, it is difficult to distinguish the residual liquid. Both values of the primary and secondary dendrite arm spacing for each welding speed are listed in Table 2.

Fig. 1.

Liquid Sn quenched microstructure during LBW at welding speed of 0.2 m/min.

Table 2 Calculation conditions.

Welding speed, m/min	0.2		1.0	1.5
Cooling rate,°C/s	745	848	2665	2896
Temperature gradient,°C/mm	223	254	316	342
Temperature measurement direction	Along centerline of molten pool	Along opened crack
Primary dendrite arm spacing, μm	10.0		6.0	5.7
Secondary dendrite arm spacing, μm	2.31		1.85	1.85

2.4. Measurement of Cooling Rate and Temperature Gradient during LBW

The solidification cracking phenomenon can be understood by observing the fracture surface of the solidification crack, which can be obtained by laser Trans-Varestraint test.²⁵⁾ Therefore, to predict the matching of the residual liquid distribution to that of the fracture surface, temperature conditions, such as the temperature gradient and cooling rate, need to be measured along the crack direction. In this case, the temperature gradient and cooling rate were measured directly using a 2D temperature distribution measured by in-situ observation.¹⁾ Figure 2 shows the method used to measure the temperature along solidification crack during LBW at 1.0 m/min. Figure 2(a) shows the solidification crack distribution at the rear center of the molten pool after the laser Trans-Varestraint test.²⁵⁾ The target crack for temperature measurement is marked by a circle. Figure 2(b) shows an image of the 2D temperature distribution along this target crack, which is expressed as a line. The temperature gradient along this crack is measured using only one 2D temperature distribution image,¹⁾ as is shown in Fig. 2(c). The cooling rate is measured at different times by using a series of images of the 2D temperature distribution, as shown in Fig. 2(d). Finally, the cooling rate and temperature gradient for each welding speed are listed in Table 2.

Fig. 2.

Measurement of temperature gradient and cooling rate along solidification crack during LBW at welding speed of 1.0 m/min. (Online version in color.)

2.5. Calculation Conditions

All of the elements should be used during calculation in principle, however, the preliminary calculation revealed that a low content induced an instabilities of calculation and it was impossible to finish the solidification due to the divergence of calculation. Therefore, elemental P, S, and Co were removed in the study. Two nucleation grains are set at both corners of a domain area with the measured primary dendrite arm spacing in Table 2. Because of the small secondary dendrite arm spacing with a higher cooling rate in Table 2, the cell dimension is set to 0.06 μm to provide sufficient resolution and to enhance calculation efficiency, and the interface thickness is set to 3 cells. During calculation, the interfacial energy was 0.3 J/m² from the literature,¹⁹⁾ and the anisotropy of interfacial stiffness was 0.10 because prior research revealed a good simulation result using this value.¹⁷⁾ The interfacial mobility and anisotropy of interfacial mobility as the calculation parameters were varied from 2.5×10⁻¹⁰ to 6.0×10⁻⁹ m⁴/J/s and from 0.005 to 0.10, respectively, because these two parameters depend on the cooling rate. The temperature along the centerline of the molten pool in Table 2 was used to evaluate the effect of calculation parameters on the secondary dendrite arm spacing and the residual liquid distribution. The temperature along the opened crack in Table 2 was used to predict the morphology and distribution of the residual liquid metal, and to verify the results with those from the experiments at each cooling rate. The liquidus temperature is 1385°C and the initial temperature was set to 1395°C to initiate calculations in a stable environment.

2.6. Comparison of Calculations with Fracture Surface of the Solidification Crack

Figure 3 shows a schematic illustration of the solidification sequence and calculated result. Blue and yellow represent the liquid and solid γ-Fe phases, respectively. According to solidification theory, a dendrite morphology (Type D) can transform into Type D-F first and then form a flat fracture (Type F) along the fracture surface of the solidification crack. This corresponds to a solidification sequence with decreasing temperature.³⁾ The critical temperature between Type D-F and Type F is defined as T_B, as shown in Fig. 3(a). In Fig. 3(b), from the primary dendrite tip at liquidus temperature T_L to the first point of the bridging of secondary dendrite arms at temperature T_B, the residual liquid connects directly with the molten pool and the liquid can move freely. Thus, this region is regarded as the “Residual liquid that connects with the molten pool region” expressed as L_P (T_L to T_B). With falling temperature from T_B to the last residual liquid at the solidus temperature T_S, the bridging becomes more significant and the residual liquid distributes as a film or a dot. Therefore, this region is termed the “Residual liquid film-dot region” and is expressed as L_FD (T_B to T_S). The mushy zone from T_L to T_S is expressed as L_M (T_L to T_S). Because the critical temperature T_B in the solidification sequence corresponds to those of the fracture morphology and calculated result, the lengths L_P and L_FD are equal to those of the region from Type D to Type D-F and the region of Type F, respectively.

Fig. 3.

Schematic illustration of (a) solidification sequence and (b) calculated result. (Online version in color.)

To verify the calculated results, the predicted residual liquid distribution was compared with the fracture surface morphology of the solidification crack. The fracture surfaces produced by the Trans-Varestraint test²⁵⁾ were studied using scanning electron microscopy. Figure 4 shows an example of the fracture surface of the solidification crack at welding speed of 0.2 m/min and a cooling rate of 848°C/s. The contour line of the crack is given by a red curve. The left side of the fracture surface is near the molten pool at the liquidus temperature T_L. Theoretically, the start hollow is the same as the first point of the bridging at temperature T_B and the end hollow means the finish of solidification at temperature T_S. Therefore, the regions from T_L to T_B and from T_B to T_S in the fracture surface correspond to L_P and L_FD in the calculated result. Moreover, the total solidification crack length is the same as that of L_M. To predict the real residual liquid distribution, the length of each region should be verified with that of the fracture surface.

Fig. 4.

Fracture surface at welding speed of 0.2 m/min and cooling rate of 848°C/s. (Online version in color.)

3. Results

3.1. Verification with Secondary Dendrite Arm Spacing and Primary Dendrite Tip Radius

To verify the calculated MPFM results, the secondary dendrite arm spacing and primary dendrite tip radius were compared with those from the experiments and the KGT modeling,^18,19) respectively. Figure 5 shows an example of the phase field distribution calculated using interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and an anisotropy of interfacial mobility of 0.10 at cooling rate of 745°C/s. Because of its long image length, the image was separated into several parts that are connected by an arrow. The blue and yellow represent 100% liquid and solid γ-Fe, respectively. The color between blue and yellow expresses the interface and fraction of liquid or solid. In the magnified image a) and b), as is mentioned, the temperature T_B and T_S can be decided at the first point of the bridging of the secondary dendrite arm and the last residual liquid metal, respectively. The secondary dendrite arm spacing and primary dendrite tip radius in this calculation are measured directly using the image such as in Fig. 5. Figure 6 compares the secondary dendrite arm spacing and primary dendrite tip radius using different interfacial mobilities and anisotropy of interfacial mobility of 0.10 at cooling rate of 745°C/s. The secondary dendrite arm spacing and primary dendrite tip radius are nearly constant for different interfacial mobilities. For the interfacial mobility from 5.0×10⁻¹⁰ to 2.0×10⁻⁹ m⁴/J/s, the secondary dendrite arm spacing and primary dendrite tip radius are both close to the experimental values in Fig. 1 and the KGT modeling results.

Fig. 5.

Phase field distribution calculated using interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and anisotropy of interfacial mobility of 0.10 at cooling rate of 745°C/s. (Online version in color.)

Fig. 6.

Comparison of secondary dendrite arm spacing and primary dendrite tip radius using different interfacial mobilities and anisotropy of interfacial mobility of 0.10 at cooling rate of 745°C/s. (Online version in color.)

Figure 7 shows the phase field distributions using interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and different anisotropies of interfacial mobility at cooling rate of 745°C/s. With increasing anisotropy of interfacial mobility, the secondary dendrite arm seems to become smaller and disappears gradually. Figure 8 shows the secondary dendrite arm spacing and primary dendrite tip radius using interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and different anisotropies of interfacial mobility at cooling rate of 745°C/s. The secondary dendrite arm spacing and primary dendrite tip radius tend to decrease with increasing anisotropy of interfacial mobility. However, above 0.10, the secondary dendrite arm disappears, as shown in Fig. 7(c). The secondary dendrite arm spacing and primary dendrite tip radius are both close to the experimental values in Fig. 1 and the KGT modeling only if the anisotropy of interfacial mobility is ~0.10. These tendencies prove that the calculated result agrees with that of not only the experiment but also the solidification theory by adjusting appropriate calculation parameters within these ranges at a cooling rate of 745°C/s. Thus, it is possible to predict the residual liquid distribution using a MPFM with a relatively high credibility under corresponding conditions for each cooling rate.

Fig. 7.

Phase field distributions calculated using interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and different anisotropies of interfacial mobility at cooling rate of 745°C/s. (Online version in color.)

Fig. 8.

Comparison of secondary dendrite arm spacing and primary dendrite tip radius using interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and different anisotropies of interfacial mobility at cooling rate of 745°C/s. (Online version in color.)

3.2. Effect of Calculation Parameters on Residual Liquid Distribution

Figure 9 shows the effect of interfacial mobility and anisotropy of interfacial mobility on the length of residual liquid region at cooling rate of 745°C/s. Figure 9(a) shows the effect of interfacial mobility on the length of the residual liquid region quantitatively. The lengths of L_P, L_FD and L_M tend to increase with increasing interfacial mobility. Figure 9(b) shows the effect of anisotropy of the interfacial mobility on the length of the residual liquid region quantitatively. With an increase in anisotropy of interfacial mobility from 0.05 to 0.10, the length of L_P increases; however, the lengths of L_FD and L_M tend to decrease. In addition, the length of each residual liquid region is nearly constant when the anisotropy of the interfacial mobility is greater than 0.10. Similar results are obtained at higher cooling rate.

Fig. 9.

Effect of interfacial mobility and anisotropy of interfacial mobility on the length of residual liquid region at cooling rate of 745°C/s. (Online version in color.)

3.3 Comparison of Residual Liquid Distribution for Different Cooling Rates

After the above verification and investigation of the effect of calculation parameters on secondary dendrite arm spacing, primary dendrite tip radius, and residual liquid distribution, it is reasonable to predict the residual liquid distribution along the opened crack. Figure 10 shows the length of residual liquid region with comparison to the length of the fracture surface in Fig. 4 at cooling rate of 848°C/s. According to the above investigations, the length of each residual liquid region is constant when the anisotropy of the interfacial mobility is greater than 0.10, thus, the calculated results are obtained first by adjusting the interfacial mobility and by setting anisotropies of interfacial mobility to 0.10, as shown in Fig. 10(a). When the interfacial mobility is 1.0×10⁻⁹ m⁴/J/s, the lengths of L_P and L_FD in the calculated result are longer and shorter than the experimental values, respectively. As mentioned previously, the lengths of L_P and L_FD tend to decrease and increase with decreasing anisotropy of interfacial mobility from 0.10 to 0.05. Therefore, the matching residual liquid distribution can be obtained only if the anisotropy of interfacial mobility is decreased and the interfacial mobility is set to 1.0×10⁻⁹ m⁴/J/s, as shown in Fig. 10(b). Finally, the length of each residual liquid region is nearly the same as that of the fracture surface of the solidification crack in Fig. 4 using an interfacial mobility of 1.0×10⁻⁹ m⁴/J/s and an anisotropy of interfacial mobility of 0.09.

Fig. 10.

Length of residual liquid region with comparison to the length of fracture surface at cooling rate of 848°C/s by adjusting parameters.

Figure 11 compares the length of residual liquid region between calculation and crack at different cooling rates by adjusting interfacial mobility and anisotropy of interfacial mobility. The length of each region in the calculated result is nearly the same as that of the fracture surface of the solidification crack. Therefore, the residual liquid distribution can be predicted by verification with the experimental result and by optimizing the calculation parameters for each cooling rate.

Fig. 11.

Comparison of length of residual liquid region between calculation and crack at different cooling rates.

The distribution of solute concentration is also investigated after verifying the length of residual liquid region. Figure 12 shows the distributions of Cr and Ni for each cooling rate. In order to understand the distribution clearly, only the images near the dendrite tip and temperature T_B are magnified and employed. Cr segregates in the liquid phase along the solid-liquid interface due to less than 1.0 of the partition coefficient. Inversely, the content of Ni in the solid phase is higher than that in the liquid phase due to more than 1.0 of the partition coefficient. This proves that the distribution of solute concentration can be calculated using MPFM. The result indicates that the predicted solidification phenomenon has a high credibility.

Fig. 12.

Distributions of Cr and Ni for each cooling rate, (a) 848°C/s, (b) 2665°C/s and (c) 2896°C/s. (Online version in color.)

4. Discussion

Interfacial mobility corresponds to an interfacial movement velocity that is attributed to driving force or curvature, which means that dendrite growth rate is directly proportional to interfacial mobility. The high dendrite growth rate can cause an increase in the length of the total mushy zone during solidification, and the length of the other residual liquid region. Therefore, the length of each residual liquid region tends to increase with an increase in interfacial mobility, as shown in Fig. 9(a). In theory, a high cooling rate can induce a high dendrite growth rate. Therefore, the interfacial mobility seems to correspond to the cooling rate. Figure 13 shows the effect of cooling rate on interfacial mobility and anisotropy of interfacial mobility by combining data from this work after verification with experimental work and previous research data.^17,21) The red symbols represent the results from this work and the white symbols are the reference values for type-304 stainless steel. In Fig. 13(a), the value of the interfacial mobility tends to increase with an increase in cooling rate during the calculation, which is the same as the above discussion.

Fig. 13.

Effect of cooling rate on interfacial mobility and anisotropy of interfacial mobility. (Online version in color.)

Anisotropy presents the difference in the degree of interfacial energy, interfacial thickness, and interfacial mobility in different directions during solidification, which causes a variation in theoretical dendrite morphology. In theory, the dendrite morphology depends strongly on interfacial mobility anisotropy at high cooling rate.²⁶⁾ Figure 14 shows a schematic illustration of the calculated results using different anisotropies of interfacial mobility based on the results in Figs. 5 and 7. When the anisotropy of the interfacial mobility decreases to a low value (such as 0.05), the extent of difference in interfacial mobility between the preferred and other growth directions decreases. The dendrite tends to grow in an isotropic manner, thus, the secondary dendrite arm becomes more significant and the primary dendrite tip radius becomes relatively large, as shown in Figs. 14(a) and 7(a). The growth of the secondary dendrite arm is conducive to promoting bridging and saving residual liquid metal in the L_FD during solidification. This results in an increase in T_B and a decrease in T_S, as shown in Figs. 14 and 7. Therefore, the above phenomenon leads to a decrease in L_P but an increase in L_FD and L_M with a decrease in interfacial mobility anisotropy from 0.10 to 0.05, as shown in Fig. 9(b). When the anisotropy of interfacial mobility has an intermediate value (such as from 0.075 to 0.10), this difference enhances the interfacial mobility between the preferred and other growth directions. Therefore, the secondary dendrite arm and primary dendrite tip radius become smaller, as shown in Figs. 14(b), 7(b), and 5. When the anisotropy of interfacial mobility is high (greater than 0.10), the columnar with the similar morphology occurs during growth, as shown in Figs. 14(c) and 7(c), resulting in the unchanged residual liquid region distribution, as shown in Fig. 9(b).

Fig. 14.

Schematic illustration of calculated results using different anisotropies of interfacial mobility, (a) low value, (b) middle value and (c) high value. (Online version in color.)

L_P tends to be shorter at higher cooling rates from those of the fracture surface. To obtain the correct residual liquid distribution, L_P must be shortened by decreasing the anisotropy of interfacial mobility. Therefore, the value of the anisotropy of interfacial mobility seems to decrease with an increase in cooling rate, as shown in Fig. 13(b). Therefore, Fig. 13 suggests a fit of reasonable calculation parameters to predict the real solidification phenomenon based on the above result.

5. Conclusion

The morphology and distribution of residual liquid in the mushy zone at different cooling rates during LBW were predicted using multi-phase field modeling and were verified experimentally. The results are as follows:

(1) The secondary dendrite arm spacing and primary dendrite tip radius of calculation obtained by adjusting the calculation parameters, such as the interfacial mobility and the anisotropy of interfacial mobility, agreed well with those of the liquid Sn quenched microstructure and KGT modeling.

(2) The residual liquid distribution could be predicted by verification with that of the solidification crack fracture surface and optimization of the calculation parameters at high cooling rates.

(3) The secondary dendrite arm spacing and primary dendrite tip radius were nearly constant with an increase in interfacial mobility. Moreover, the length of residual liquid distribution increased. With an increase in anisotropy of interfacial mobility, the secondary dendrite arm spacing and primary dendrite tip radius tended to decrease. The lengths of L_P increased and were then kept constant. However, the lengths of L_FD and L_M decreased and were kept the same.

(4) The variation in interfacial mobility and anisotropy of interfacial mobility tended to increase and decrease with an increase in cooling rate, respectively. This law can be applied to fit suitable calculation parameters to predict real solidification phenomena at various cooling rates.

References

• 1)   D.  Wang,  K.  Kadoi,  K.  Shinozaki and  M.  Yamamoto: ISIJ Int., (2016), DOI: http://dx.doi.org/10.2355/isijinternational.ISIJINT-2016-302.
• 2)   K.  Kadoi,  A.  Fujinaga,  M.  Yamamoto and  K.  Shinozaki: Weld. World, 57 (2013), 383.
• 3)   J. C.  Lippold and  D. J.  Kotecki: Welding Metallurgy and Weldability of Stainless Steels, John Wiley & Sons, Hoboken, NJ, (2005), 186.
• 4)   B.  Liu and  Q.  Xu: Tsinghua Sci. Technol., 9 (2004), 497.
• 5)   J.  Gao and  R. G.  Thompson: Pro. the Sixth International Symposium on Superalloys 718, 625, 706 and Various Derivatives, Minerals, Metals and Materials Society, Warrendale, PA, (1997), 77.
• 6)   P.  Zhu and  R. W.  Smith: Acta Metall. Mater., 40 (1992), 683.
• 7)   X.  Liu,  Q.  Xu,  T.  Jing and  B.  Liu: Trans. Nonferrous Met. Soc. China, 19 (2009), 422.
• 8)   H. W.  Hesselbarth and  I. R.  Gobel: Acta Metall. Mater., 39 (1991), 2135.
• 9)   T.  Biben: Eur. J. Phys., 26 (2005), 47.
• 10)   R. S.  Qin and  E. R.  Wallach: Acta Mater., 51 (2003), 6199.
• 11)   S. G.  Kim,  W. T.  Kim and  T.  Suzuki: Phys. Rev. E., 58 (1998), 3316.
• 12)   S. G.  Kim,  W. T.  Kim and  T.  Suzuki: Phys. Rev. E., 60 (1999), 7186.
• 13)   T.  Suzuki,  M.  Ode,  S. G.  Kim and  W. T.  Kim: J. Cryst. Growth, 237–239 (2002), 125.
• 14)   D. J.  Seol,  K. H.  Oh,  J. W.  Cho,  J.  Lee and  U.  Yoon: Acta Mater., 50 (2002), 2259.
• 15)   Y.  Xie,  H.  Dong and  J.  Dantzig: ISIJ Int., 54 (2014), 430.
• 16)   W. J.  Zheng,  Z. B.  Dong,  Y. H.  Wei,  K. J.  Song,  J. L.  Guo and  Y.  Wang: Comput. Mater. Sci., 82 (2014), 525.
• 17)   S.  Fukumoto and  I.  Hiroshige: Q. J. Jpn. Weld Soc., 29 (2011), 197.
• 18)   W.  Kurz,  B.  Giovanola and  R.  Trivedi: Acta Metall., 34 (1986), 823.
• 19)   W.  Kurz and  D. J.  Fisher: Fundamentals of solidification, Trans Tech Publications, Aedermannsdorf, (1989), 293.
• 20)   I.  Steinbach,  F.  Pezzolla,  B.  Nestler,  M.  Seebelberg,  R.  Prieler,  G. J.  Schmitz and  J. L. L.  Rezende: Phys. D, 94 (1996), 135.
• 21)   S.  Fukumoto and  S.  Nomoto: J. Jpn. Inst. Met., 73 (2009), 502.
• 22)   J.  Eiken,  B.  Bottger and  I.  Steinbach: Phys. Rev. E, 73 (2006), 066122-1.
• 23)   J. J.  Eggleston,  G. B.  McFadden and  P. W.  Voorhees: Phys. D, 150 (2001), 91.
• 24)  MICRESS group: User Guide Version 6.1 Volume 2: Running MICRESS, MICRESS group, Aachen, Germany, (2013), 66.
• 25)   D.  Wang,  S.  Sakoda,  K.  Kadoi,  K.  Shinozaki and  M.  Yamamoto: Q. J. Jpn. Weld Soc., 33 (2015), 39.
• 26)   E. A.  Brener: Sov. Phys. JETP, 69 (1989), 133.