Analytical Prediction of Molten Pool Dimensions in Powder Bed Fusion Considering Process Conditions-Dependent Laser Absorptivity

Abstract

This research proposes an analytical method for the prediction of molten pool size in laser-based powder bed fusion (LPBF) additive manufacturing with the consideration of process conditions-dependent absorptivity. Under different process conditions, the melting modes in LPBF are different, which induces the difference in laser absorptivity. An empirical model of absorptivity was used to calculate the laser absorptivity under various process conditions. An analytical point-moving heat source model was employed to calculate the temperature distribution of the build-in LPBF, with absorptivity, material properties, and process conditions as inputs. The molten pool width, length, and depth were determined by comparing the predicted temperature profile with the melting temperature of the material. To validate the proposed method, the predicted molten pool width, and depth of Ti6Al4V were compared with the reported experimental measurements under various process conditions. The predicted molten pool widths were very close to the measured results, and the predictions of molten pool depth were also acceptable. The computational time of the presented model is less than 200s, which shows better computational efficiency than most methods based on numerical iterations, such as the finite element method (FEM). The sensitivity of molten pool width and depth to normalized enthalpy w also discussed. The presented analytical method can be a potential tool for the research of molten pool size and related defects in LPBF.

1. Introduction

In recent years, laser powder bed fusion (LPBF) additive manufacturing has attracted more and more interest from academia and industries due to its superior advantage in producing complex metallic parts than traditional manufacturing methods [1,2,3,4,5]. In LPBF, metallic parts are fabricated by melting metal powders in a layer-by-layer style [6]. The molten pool dimensions in the manufacturing process are important factors that affect the quality of the final products. The common defects, such as surface roughness, lack-of-fusion porosity, balling, and keyholing are all related to the molten pool behavior in LPBF [1,7,8,9,10]. Therefore, a good prediction model for the molten pool dimensions is necessary for quality control in LPBF.

It was observed that there are three melting modes in the powder bed fusion process, which are the conduction mode at the low energy density regime, transition mode at the medium energy density regime, and keyhole mode at the high energy density regime [11,12,13]. Under different process conditions of LPBF, the laser power absorptivity is different due to the difference in melting modes. In the conduction melting mode, the vapor depression depth and molten pool depth are small, most laser power is reflected by the molten pool surface to the ambient, which induces low laser absorptivity in this process regime. In the transition mode, the vapor depression depth increases with the energy density, the laser has more reflections in the vapor hole, which induces an increase in laser absorptivity. When it comes to the keyhole mode melting regime, the absorptivity goes to the highest level and will not change much with the process conditions [11,14,15,16]. Laser absorptivity has a significant influence on the energy input to the material in LPBF. Thus, the consideration of the difference in laser absorptivity is necessary for the prediction of molten pool under different process conditions. This paper focuses on the laser absorptivity in the transition mode and keyhole mode.

Researchers have developed various methods to study the laser absorptivity and molten pool behavior in LPBF [17,18,19]. Plenty of experimental techniques have been used to measure the absorptivity and investigate the characteristics of the molten pool. Trapp et al. [10] employed a direct calorimetric technique to measure the laser absorptivity of 316L stainless steel substrate which is coated by a powder layer. Ye et al. [20] used the same experimental technique to study the laser absorptivity of Ti6Al4V powders. The sensitivity of absorptivity to laser beam size, the thickness of powder layers, laser power, and scanning speed were investigated. Rubenchik et al. [21] developed a calorimetric apparatus to investigate the laser absorptivity of Ti6Al4V and stainless-steel powders. The effect of temperature on absorptivity was studied. Dilip et al. [22] employed optical micrographs to investigate the molten pool shapes and dimensions of Ti6Al4V in single-track experiments of the selective laser melting process. Gunenthiram et al. [23] used a SA2 Photron–Fastcam high-speed camera to monitor the molten pool behavior in LPBF. Zhao et al. [24] and Cunningham et al. [1] employed the high-speed x-ray imaging technique to observe the molten pool shapes during the powder bed fusion process and measure the molten pool dimensions. These experimental investigations have made significant contributions to the research of the powder bed fusion process. However, the complex experimental procedure and high cost of experimental devices hinder the wide adoption of these techniques.

Numerical methods have been an alternative for the experimental investigation for the study of absorptivity, temperature profile, and evolution of molten pools in LPBF. Boley et al. [25] carried out ray-tracing simulations to compute the laser absorptivity of powder materials for additive manufacturing processes. Karayagiz et al. [26] developed a finite element model to simulate the thermal history in LPBF. A constant metal powder absorptivity was employed in the simulation. Vastola et al. [27] also employed a finite element method to calculate the temperature profile of the part in additive manufacturing, considering temperature-dependent material properties. The molten pool dimensions and cross-sectional area were obtained through the FEM simulations. Lee et al. [28] developed a transient model to compute the temperature distribution and molten pool behavior in additive manufacturing. The powder packing pattern was considered by a discrete element method. Khairallah et al. [29] employed a multi-physics model to simulate the molten pool behavior and defect formation in additive manufacturing processes. Foroozmehr et al. [30] proposed a finite element model for the prediction of temperature profiles and molten pool geometries in parts fabricated by the selective laser melting process. The Optical Penetration Depth of laser power was taken into consideration in the model. Lee et al. [31] developed a new heat source model for the prediction of molten pool size through a finite element software. A relationship between laser absorptivity and molten pool depth was used in the simulations. Although iteration-based numerical methods can save the cost of experimental equipment, the computational cost of these methods is still too high to be widely used by academia and industry.

To save computational resources, researchers proposed some efficient analytical modeling methods to study the LPBF process, based on some reasonable assumptions. Fergani et al. [32] developed an analytical method to study the residual stress in the selective laser melting process. Ji et al. [33] predicted the grain size characteristics in metal additive manufacturing using an analytical strategy. Ning et al. [34] proposed a complete analytical method to compute the part porosity in LPBF induced by lack-of-fusion, and the effect of powder packing density was considered. Wang et al. [7] developed a simple analytical expression to estimate the surface roughness of parts built by selective laser melting, with the assumption of a circular molten pool cross-section above the substrate. Ning et al. [35] proposed an analytical temperature prediction method based on the point-moving heat source model and a heat loss solution. The heat loss through part boundaries was considered in that model. Tang et al. [36] and Seidel et al. [37] employed the Rosenthal equation to predict the temperature profiles and molten pool dimensions in LPBF, with absorptivity as constant values. Promoppatum et al. [38] also used the Rosenthal equation to obtain the molten pool size. The sensitivity analysis of predicted results to laser absorption was conducted in that paper. However, there is no reported model in the literature which takes into account the relationship between laser absorptivity and process conditions in the prediction of temperature distribution and molten pool shapes. The coefficient of laser absorption will significantly be affected by the process conditions in the laser powder bed fusion process [39,40]. The consideration of process conditions-dependent laser absorptivity will simulate the melting behavior in LPBF better and thus increase the predictive accuracy of the computational models for temperature distribution and molten pool geometries.

In this research, the molten pool dimensions of Ti6Al4V in powder bed fusion additive manufacturing were predicted by an analytical thermal model, with the consideration of the difference in laser absorptivity under different process conditions. A scaling law-based empirical model for absorptivity was employed to compute the different laser absorptivity in LPBF. A point moving heat source model with a closed-form solution was used to compute the temperature profiles in single tracks. The absorptivity, thermal properties of the material, and process conditions were the inputs for the analytical thermal model. By comparing the calculated temperature distribution with the melting temperature, the molten pool shapes were determined. The predicted molten pool dimensions were compared with experimental measurements of Ti6Al4V to validate the presented analytical method. The computational time was recorded. The sensitivity of molten pool width and depth to normalized enthalpy were studied.

2. Analytical Modeling

This study presents an analytical strategy to predict the temperature profiles and estimate the molten pool dimensions in the powder bed fusion process, with the consideration of process conditions-dependent absorptivity. An empirical model was employed to correlate the process conditions with the laser absorptivity of powder materials. A closed-form heat source model was used to compute the temperature distribution, based on the information on material properties, process conditions, and absorptivity. The molten pool geometries were obtained by comparing the melting temperature of the material with the calculated temperature profiles.

The empirical model for the calculation of absorptivity was proposed by Ye et al. in [20], which was obtained by scaling law and curve fitting of plenty of experimental measurements of absorptivity of various materials, including Ti6Al4V, 316L stainless steel, and Inconel 625. This model captures the change of laser absorptivity in the transition-mode regime and keyhole melting regime [20]. It can be expressed as:

$$A_e=0.70\left(1-e^{-0.66{\beta }_{A_m{L}_{th}^{*}}}\right)$$

(1)

where, $A_e$ represents the effective absorptivity, which will be used as the input absorptivity for the thermal model. ${\beta }_{A_m}=\frac{A_{m}P}{\pi \rho c{T}_m\sqrt{Du{a}^3}}$ represents the normalized enthalpy when the absorptivity is the minimum of absorptivity $(A_m)$ measured in the transition regime. In this equation, $P$ is the laser power, $u$ is the scanning speed, $a$ denotes the laser beam radius, $\rho , c T_m$ are the material properties, density, specific heat capacity and melting temperature, respectively. $L_{th}^{*}=\left(\sqrt{Da/u}\right)/a$ represents the normalized thermal diffusion length [20].

The analytical thermal model to calculate the temperature distribution is a point moving heat source model, which was derived by Carslaw et al. in [41] based on the assumptions of semi-infinite medium, thermal conduction, and temperature-independent material properties. It should be noted that there will be heat loss at part boundaries in LPBF through thermal conduction, convection, and radiation [42]. In this thermal model, the boundary heat loss is not considered. The thermal model can be expressed as:

$${\theta }_{laser}\left(x,y,z\right)=\frac{PA}{2\pi kR\left(T_m-T_0\right)}exp\left(\frac{-u\left(R+x\right)}{2\kappa }\right)$$

(2)

In this equation, ${\theta }_{laser}=\frac{T-T_0}{T_m-T_0}$ is the dimensionless temperature, $T_0$ and $T_m$ denote the melting temperature and initial temperature (room temperature or preheating temperature), respectively. $P$ and $u$ represent the laser power and moving speed. $A$ represents the laser absorptivity of the material. $\kappa =\frac{k}{\rho c}$ is the thermal diffusivity obtained from thermal conductivity $k$, density $\rho$ and heat capacity $c$. $R=\sqrt{x^2+y^2+z^2}$ is the distance of the calculated point from the heat source, $\left(x,y,z\right)$ represent the coordinate in the calculation domain.

3. Experimental Validation and Discussion

In this research, an analytical modeling strategy was developed for the molten pool geometries prediction in LPBF. The process conditions-dependent laser absorptivity was calculated by an empirical model. The molten pool shapes were determined by comparing melting temperature with the temperature field predicted by a closed-form point moving heat source model. Ti6Al4V was employed as the material to validate the prediction accuracy of the presented analytical method. The experimental measurements of the molten pool size of Ti6Al4V in single-track experiments were used for validation, and were reported in [22]. The process conditions and experimental data of molten pool size are shown in

Table 1. Comparison between experimental [22] and predicted molten pool size of Ti6Al4V.

Case Number	Power (W)	Scanning Speed (mm/s)	Normalized Enthalpy	Calculated Absorptivity	Measured Results		Predicted Results
					Depth (µm)	Width (µm)	Depth (µm)	Width (µm)	Length (µm)
1	100	500	4.6	0.57	43.6	117.9	54.5	105.5	180.9
2	100	750	3.7	0.48	31.9	98.0	42.4	85.4	145.7
3	150	500	6.9	0.65	100.9	145.0	72.7	145.7	291.5
4	150	750	5.6	0.57	72.0	134.4	58.6	115.6	251.3
5	150	1000	4.9	0.51	52.7	115.3	46.5	95.5	216.1
6	150	1200	4.4	0.46	49.2	108.1	42.4	85.4	201.0
7	195	500	8.9	0.68	174.9	192.8	86.9	175.9	386.9
8	195	750	7.3	0.62	110.5	160.6	68.7	135.7	351.8
9	195	1000	6.3	0.57	81.1	129.7	58.6	115.6	316.6
10	195	1200	5.8	0.53	63.9	122.5	50.5	95.5	286.4

. It should be noted that the molten pool dimensions were measured through image analysis of the optical micrographs of the molten pool cross-sections.

To calculate the absorptivity under different process conditions, the material properties of Ti6Al4V are needed as inputs, which are shown in

Table 2. Material properties of Ti6Al4V [34,44,45].

Name	Symbol	Value	Unit
Density	ρ	4198	kg/m3
Thermal conductivity	k	27	W/(m·℃)
Specific heat	c	750	J/(kg·℃)
Room temperature	T0	20	℃
Boiling temperature	Tb	3042	℃
Melting temperature	Tm	1655	℃

. These material properties are assumed to be temperature-independent, which are chosen as the values at melting temperature $\left(T_m\right)$ of Ti6Al4V. The minimal absorptivity ($A_m$) of Ti6Al4V in the transition regime from conduction mode to keyhole mode was measured as 0.45 [43]. The laser beam radius is 50 $\mu m$ [22]. The calculated absorptivity under various process conditions is also shown in Table 1. It should be noted that the initial temperature for the LPBF process in this study is room temperature. The proposed model can consider the preheating effect by changing the initial temperature to the preheating temperature of the LPBF process.

With the information on thermal properties, process conditions, and absorptivity, the point moving heat source model was employed to compute the temperature distribution and obtain the estimation of molten pool dimensions. Figure 1 shows the temperature profiles and molten pools of the central cross-section in the x, z plane (x axis is parallel to the scanning direction). The predicted molten pool width and depth with different absorptivity are shown in Table 1. The molten pool dimensions increase with the laser power and decrease with the scanning speed. To demonstrate the presented method, these predicted results were compared with the experimental data, as shown in Figure 2. It can be observed that the predictions of molten pool width are very close to experimental data, and the predictions for depth have an acceptable agreement with the experimental results, which show the good prediction accuracy of the proposed analytical strategy.

The normalized enthalpy $\beta =\frac{AP}{\pi \rho c{T}_m\sqrt{Du{a}^3}}$ under various process conditions were calculated and shown in Table 1. The sensitivity of the molten pool width and depth to the normalized enthalpy is shown in Figure 3. It can be observed that both the experimental [22] and analytical results have a nearly linear relationship with the normalized enthalpy. An empirical formula obtained from the linear relationships may be an acceptable alternative to predict the molten pool dimensions [46]. A criterion derived by King et al. in [12] was employed to calculate the threshold for the occurrence of the keyhole melting mode, which can be expressed as $\beta >\frac{\pi T_b}{T_m}$, where $\beta$ is the normalized enthalpy, $T_b$ and $T_m$ are the boiling point and the melting point of the material, respectively. When the normalized enthalpy is over 5.4, the melting mode of Ti6Al4V will become keyhole mode, which is also shown in Figure 3. It can be observed that the predictions for molten pool depth are closer to the experimental results when the normalized enthalpy is lower than the threshold. When the process conditions enter the keyhole-mode melting regime, the predictions of molten pool depth cannot capture the variation of experimental results very well. This is mainly because the point moving heat source model only considers the thermal conduction [41], however, in the keyhole melting regime, the thermal convection plays an important role [11]. In addition, the predicted results with uniform absorptivity of 0.45 are shown in Figure 3 for comparison, which indicates that considering the variation of laser absorptivity can enhance the prediction accuracy. The computational time for the presented method is less than 200 s, which can save plenty of computational resources than finite element-based numerical methods.

By virtue of the acceptable prediction accuracy and high computational efficiency, the presented analytical method can work as an efficient tool to predict the molten pool shapes and study the related defects in the powder bed fusion processes. In the future, the effects of temperature-dependent material properties and the thermal convection in keyhole regime can be considered to make the predictions more accurate.

4. Conclusions

In this study, an analytical modeling method was proposed to predict the molten pool dimensions in laser powder bed fusion additive manufacturing, with the consideration of process conditions-dependent absorptivity. The absorptivity under different process conditions were calculated by an empirical model of laser absorptivity. The temperature profiles in single tracks were calculated by a point moving heat source model with the laser absorptivity, material thermal properties, and process conditions as inputs. After comparing the predicted temperature distribution and melting temperature of the material, the molten pool dimensions were obtained. Ti6Al4V was used as the material to validate the presented analytical method. The predictions of molten pool width and depth were compared to the experimental measurements of Ti6Al4V. The predicted results show good agreement with the experimental data, which indicates the acceptable prediction accuracy of the presented model. In addition, the predicted results were compared against the predictions with constant absorptivity, which indicates that the consideration of different absorptivity can enhance the accuracy in the molten pool predictions. The computational time of the proposed method is less than 200 s, which is much less than that of FEM-based numerical methods. The sensitivity analysis shows that the molten pool with and depth have a nearly linear relationship with normalized enthalpy. The presented analytical method can predict acceptable results and save many computational resources, which makes it a potential tool for the study of molten pools and defects in LPBF.