《1.Introduction》

1.Introduction

Electron beam selective melting (EBSM) is a promising additive manufacturing (AM) technology for metallic components. It is capable of manufacturing components with complex geometry, and opens up new avenues for locally manipulating chemical compositions and mechanical properties as well. For example, Yang et al. [1] manufactured auxetic lattice structures with negative Poisson’s ratios, and Ge et al. [2–4] manufactured functionally graded Ti-TiAl materials.

There are three main fabrication procedures in EBSM [2], as shown in Fig. 1.

《Fig. 1》

Fig. 1. Experiments and models of all procedures in the EBSM process. DEM: discrete element method; PF: phase field; FVM: finite volume method.

(1)Spread one layer of powder on a preheated platform or previous layer. The layer thickness can vary for different layers. For each layer, the mixture ratio of the several different types of powder can be designed and tailored in order to allow the chemical compositions to be manipulated.

(2)Preheat the powder bed to make the powder slightly sintered. This helps prevent powder scattering, which may even lead to build failure.

(3)Selectively melt the powder bed. The beam power and scan speed are key factors that greatly influence the final part quality.

Although the basic principle of EBSM is rather straightforward, the actual processes consist of multiple physical phenomena such as powder particle  packing, heat  transfer, phase  transformation, and fluid flow, and a number of factors influence the process and fabrication quality. There are a considerable number of fundamental physical mechanisms in each fabrication procedure to be understood in order for optimal process parameters to be selected to ensure the fabrication quality. For example, the questions of how to improve the packing density of the powder bed in the powder-spreading procedure, how to achieve the optimal coalescence state of the powder bed in the preheating procedure, and how to avoid the balling effect and reduce single-track non-uniformity are all meaningful research topics.

Most previous studies focused on the melting procedure rather than on the other two procedures. Few studies have been done to  comprehensively  model  all  the  manufacturing  procedures. A few powder-scale models resolving the randomly distributed particles in  the  powder  bed have  been  developed  to investigate the melting process of individual powder particles [5–7]. Körner et al. [5] employed the rain model to generate the powder layer and the  two-dimensional  (2D)  lattice  Boltzmann  method  (LBM) to model the powder-melting process. They studied the influence of the powder layer thickness and input energy on the successive consolidation process of multiple powder layers. Khairallah et al. [6] built a meso-scopic model for selective laser melting (SLM) in order to investigate the formation  mechanism  of  pores,  spatter, and denudation in the single-track formation process using the ALE3D multi-physics code. Qiu et al. [8] used the open-source code OpenFOAM to simulate the powder-scale melt flow in the SLM process in order to study the surface structure and porosity development. These models incorporated most of the driving forces of the molten pool flow, including surface tension, the Marangoni effect, and recoil pressure. In this work, we leverage modeling and experiments to advance the understanding of physical  mechanisms  in each of the three procedures (Fig. 1). The models are introduced in Section 2, and include a powder-spreading model using the discrete element method (DEM), a phase field (PF) model of powder sintering, and a powder-melting model using the finite volume method (FVM). Section 3 presents  the experimental methods. In Section 4, experimental and simulation results for each procedure are presented and discussed. Finally, a brief summary is given in Section 5.

《2. Models》

2. Models

Note that the notations in the three subsections apply only in the respective subsections.

《2.1. Powder-spreading model》

2.1. Powder-spreading model

Spherical powder particles with diameters that follow a Gaussian distribution in the range of 30–50 μm firstly fall to the bottom under gravity to form a powder bed, covering the substrate with various thicknesses. The rake then moves from left to right to spread the powder (Fig. 1). The movement of powder particles is governed by the contact interaction and body forces.

The Hertz-Mindlin contact model is used. In a simple case where there are only two contact particles with radii R1 and R2, the contact forces in the normal and tangential directions consist of nonlinear deformation and damping, as given in Eq. (1) and Eq. (2).

where δn  and |ξt|, and |δn| and |ξt| are the relative displacements and velocities in the normal (en) and tangential (et) directions, respectively; while Kn and Kt, and γn and γt are the effective stiffnesses and damping coefficients in the normal and tangential directions, respectively, given in Eq. (3).

where E, υ, and θ are the Young’s modulus, Poisson’s ratio, and restitution coefficie.nt, respectively.

If , tangential slide will happen, and then the tangential force is the maximum friction as given in Eq. (4).

where μ is the fraction coefficient.

In addition to the contact repulsive force, the cohesive nature of fine particles exhibits an attractive force. Here we use the JohnsonKendall-Roberts (JKR) cohesion model originally implemented to incorporate the Van der Waals forces between fine and dry particles, as given in Eq. (5). The cohesion force is mainly determined by the surface energy density, Ψ, of the material.

where a is the radius of the contact area.

《2.2. Phase field model of powder sintering》

2.2. Phase field model of powder sintering

It should be emphasized that the powder-sintering mechanism during the preheating procedure is solid-state sintering driven mainly by grain boundary diffusion, rather than liquid-state sintering driven by melting and solidification. This solid-state sintering during the preheating procedure in EBSM has rarely been studied or modeled. Thus, we propose to use the PF to model the powder sintering. For brevity, we explain the method in the case of two particles, in which case two kinds of fields are used: the volume of solid, c, and the order parameter fields, η1  and η2. The value of all these fields falls between 0 and 1, as illustrated in Fig. 2. Note that on the grain boundary, c = 1.

《Fig. 2》

Fig. 2. Schematic of the fields in the PF model.

The evolution of the sintering process is driven by minimizing the total free energy, F [9]:

where v is the volume; f is the bulk free energy; and κc and κη are gradient energy parameters. Thus, we have the Cahn-Hilliard equation for the conserved field, c:

and the Allen-Cahn equation for the non-conservative order parameter field:

where M is the concentration mobility tensor; L is the order parameter scalar mobility; x is the spatial position vector; and t represents time.

The package FEniCS [10] is adopted to solve this nonlinear problem. The Newton solver and the monolithic method are applied. Since the preheating procedure for each powder layer usually takes about 20 s, a constant time-step dt = 2 × 10–4 s is adopted, which should be sufficient to resolve this solid-state sintering process. Periodic boundary conditions are applied and an implicit solver is employed.

《2.3. Hydrodynamic model of powder melting》

2.3. Hydrodynamic model of powder melting

The powder-melting process is governed by continuity (Eq. (9)), energy conservation (Eq. (10)), and momentum conservation (Eq. (11)), which are described in detail as follows:

where ρ and v denote the mass density and velocity vector of melt flow, respectively. In this study, the melted material flow is assumed to be Newtonian incompressible flow, and the mass density is set to be constant.

where k is the thermal conductivity; T is the temperature; and I = cT + (1 – fs) L is the specific internal energy, in which c and L are the specific heat and latent heat of melting, respectively, and fs is the fraction of the solid. Most importantly, q is the input heat source model, which is established based on the Monte Carlo simulations of electron-atom interactions [11]. The heat source model is material dependent and experimental setup specific. Detailed descriptions are provided in our previous papers [11–13]. The initial condition is a uniform temperature distribution of 873 K, in order to incorporate the preheating procedure. The thermal boundary conditions are surface radiation and heat loss due to surface evaporation.

 where g denotes the gravitational acceleration vector; μ is the viscosity; and p is the pressure. The buoyancy fB   is accounted for using the Boussinesq approximation (Eq. (12)).

where β is the thermal expansion coefficient; and T0  is the reference temperature.

The free surface is tracked using the volume-of-fluid (VOF) method. As given in Eq. (13), the phase fraction (F) in each cell is calculated, while the velocity (v) is obtained from the solution of the momentum equation given by Eq. (11). Based on the phase fraction, the free surfaces can be reconstructed in each time-step.

The recoil pressure, surface tension, and Marangoni effect are treated as the boundary conditions on the free surface, while the surface tension coefficient is set to be temperature dependent.

《3. Experimental methods》

3. Experimental methods

The EBSM system used in the study (Fig. 3) is open-architecture, which allows users to customize a wide variety of fabrication parameters. A detailed description can be found in a previous paper [4]. Since there are two powder tanks, the equipment is capable of manufacturing functionally graded materials by tailoring  the  mixture ratio of the two types of powder in each layer.

《Fig. 3》

Fig. 3. The in-house EBSM system. (a) Schematic; (b) photograph.

For  the  powder-spreading  experiments,  both  the  translational speed and the slope angle of the powder rake can be customized. The powder size distribution is measured using the laser particle size analyzer.

After the powder bed is applied and before the selective melting procedure, the powder layer is usually preheated using a de-focused electron beam. The purpose is to slightly sinter powder particles together in order to avoid powder “smoke.” In the post-process experimental characterization, in order to investigate the mechanism of sintering, we attached the sintered powder particles to some embedding resins at 100 °C, and then employed a focused ion beam (FIB) instrument to do the polishing. Finally, we used a scanning electron microscope (SEM) to observe the microstructures at the sintering neck at the contact point of the powder particles. Mechanical polishing is not feasible, since it may break the neck and change the microstructure.

In the powder-melting procedure, a focused electron beam  is used. The morphology of single tracks is observed under an optical microscope (OM).

《4. Results and discussion》

4. Results and discussion

《4.1. Powder spreading》

4.1. Powder spreading

A good index to evaluate the performance of powder-spreading machinery is the relative packing density of the powder bed. A more compacted powder bed is usually beneficial to the fabrication quality, which can be demonstrated by the powder-melting model, as introduced in Section 2.3.

The simulations (Fig. 4(a)) can guide the design and optimization of the powder rake (Fig. 4(b)), including the rake shape and its translational speed. The relative packing density over either a flat substrate or the fluctuating surface of previous layers can be predicted (Fig. 4(c)) and then compared with experiments (Fig. 4(d)).

《Fig. 4》

Fig. 4. Experimental and simulation results of powder spreading. (a) Simulations can guide the design and optimization of the (b) powder rake; (c) simulation and (d) experimental results of spreading a powder layer over previous layers.

Some preliminary simulation results reveal the following phenomena:

  • If the translational speed is relatively low, the rake shape does not affect the packing density, and the resultant packing density is high.
  • The packing density decreases with the increase of the rake speed, as illustrated in Fig. 5. It should be noted that the effect of rake vibration is not incorporated into the current model; however, in experiments, the vibration is influenced by the rake speed, and then in turn influences the powder spreading [4]. It should also be mentioned that the first powder layer has a lower packing density than that of the whole powder bed, since the layer thickness is only around two times the mean powder particle diameter.

《Fig. 5》

Fig. 5. Powder-spreading simulation results at various rake speeds.

《4.2. Powder sintering》

4.2. Powder sintering

The PF modeling of sintering is primarily in 2D. Two simulation cases are presented: ① two powder particles with different sizes, and ② two powder particles with similar sizes (Fig. 6). In Fig. 6(a) and Fig. 6(c), the red portion denotes the domain full of material, the blue portion denotes the domain without material, and transitional colors represent the material interfaces. Table 1 [14–16] lists some material parameters applied.

Although there are still quite a few improvements to be made to this model, the experimental observation (Fig. 6(b) and Fig. 6(d)) has been qualitatively reproduced, which demonstrates the modeling ability of the proposed method for the sintering process. It should be noted that the scales are different between the modeling and the experiment. To reach the same sintering stage, a larger scale will result in a much longer time, due to the size effect.

《Table 1》

Table 1 Material parameters applied in the powder-sintering simulation.

《Fig. 6》

Fig. 6. Experimental and simulation results of powder sintering. (a), (b) two powder particles with different sizes; (c), (d) two powder particles with similar sizes.

《4.3. Powder melting》

4.3. Powder melting

One important feature of our high-fidelity powder-melting model is the accurate implementation of the heat source model by accurately capturing and reconstructing the material surfaces employing an enhanced VOF method [17]. To illustrate this, we simulated the process of an electron beam (in which the energy is evenly distributed within the beam’s cross-section) heating a spherical powder particle on a substrate. In Fig. 7, it can be seen that the electron beam can penetrate through the edge of the particle into the substrate below, since the maximum penetration depth of the electron beam is about 16 μm. It is also noted that the subsurface regions (about 5 μm below the surface) have a higher temperature than the surface, thus perfectly incorporating the energy distribution along the penetration depth from the microscale simulation of electron-atom interactions [11,12]. Moreover, in Fig. 7(b), the red region (with a temperature higher than 1928 K) becomes smaller with the increase of distance from the center, because the energy absorptivity is higher near the center and lower near the edge, due to the influence of the incidence angle [12].

《Fig. 7》

Fig. 7. Simulation result of an electron beam heating a spherical powder particle on a substrate. (a) Schematic; (b) simulated.

The single track acts as one fundamental building unit and largely influences the final product quality, such as the surface roughness and dimensional accuracy. We employed the FVM-based high-fidelity powder-scale model to predict the detailed formation processes of single-track defects, including the balling effect and single-track non-uniformity (Fig. 8). These processes are difficult to observe in experiments; previous studies have proposed different or even conflicting explanations. The model helped clarify the underlying formation mechanisms, reveal the influence of key factors, and guide the improvement  of fabrication  quality.  More  detailed discussion and description are in our previous papers [13].

The major conclusions are:

  • The balling effect is caused by a lack of melting of the substrate under the melted particles. Driven by the surface energy, some of the melted particles merge together into isolated clusters rather than spreading over the unmelted substrate surface.
  • The single-track non-uniformity is due to the random attachment of the molten pool to the partially melted particles near the boundaries. In multiple-layer multiple-track manufacturing processes, the previous layers and tracks, as well as the ejected materials, will also influence the non-uniformity of the melt track. The detailed simulation results and discussion are out of the scope of this paper, and can be found in our previous papers [13].

《Fig. 8》

Fig. 8. Experimental and simulation results of (a) and (c) balling effect; and (b) and (d) single-track non-uniformity [13].

《5. Summary》

5. Summary

To achieve a comprehensive understanding of the physical mechanisms in the EBSM process, we leverage modeling and experiments. The models consist of ① a powder-spreading model using the DEM,② a PF model of powder sintering, and ③ a powder-melting model using the FVM. These models fully resemble the actual fabrication procedures. Preliminary simulation results agree qualitatively with experiments, and demonstrate appealing potential to shed light on the underlying mechanisms and to guide the design and optimization of experimental setup and manufacturing processes.

《Acknowledgements》

Acknowledgements

The authors at Tsinghua University acknowledge the financial support by the National Key R&D Program of China(2017YFB1103303) and the Suzhou-Tsinghua Innovation Leading Action Specials (2016SZ0216). Yongxing Shen acknowledges the financial support by the Recruitment Program of Global Experts.

《Compliance with ethics guidelines》

Compliance with ethics guidelines

Wentao Yan, Ya Qian, Weixin Ma, Bin Zhou, Yongxing Shen, and Feng Lin declare that they have no conflict of interest or financial conflicts to disclose.