1 Introduction

Laser powder bed fusion (LPBF) has become one of the most prevalent additive manufacturing technologies for producing metallic parts and has been used in many industry fields [1]. Nevertheless, there are also many scientific and engineering issues remaining unsolved. In the LPBF process, a high-energy density laser beam is used to fuse a powder bed, layer by layer. Since the powders are not firmly adhered to the substrate, it can be displaced from its initial deposition or spreading position due to various physical forces. This displacement can compromise the density of the powder bed, leading to metallurgical defects such as spatter and porosity. Such defects may deteriorate the quality of the fabricated part [2].

Powder movement is a prevalent phenomenon in the LPBF process, as corroborated by some observational experiments. It is known to be mainly driven by gas flow condition [3]. However, quantitatively identifying this metal vapor remains a challenge. Capturing the vapor plume experimentally requires an exceptionally well-designed setup and relies heavily on the use of high-resolution instruments. Bidare et al. [4] employed schlieren imaging to visualize the laser plume, subsequently identifying three distinct powder denudation regimes. However, the dynamic evolutions and relationship between the laser plume and the mechanism of powder movement were not explored. More recently, Bitharas et al. [5] combined schlieren imaging with X-ray synchrotron imaging techniques capable of simultaneously capturing the plume profile, melt pool and powder movement. Despite their observations were limited to a 2D perspective, they were the first to shed light on the intricate interactions between the metal vapor, the liquid melt pool, and the solid powder.

Utilizing physics-based numerical simulations provides a commendable avenue to quantitatively study the evolution and inherent mechanisms of melt-vapor-powder interactions. However, a majority of the existing research tend to isolate these interactions and model them independently [6,7,8,9,10,11,12,13,14,15,16] partly because the modelling of powder movement and the fluid flow are based on different categories of physics formulas. Chen et al. [17] combined these two systems, while ignoring the explicit modelling of the melting and solidification process. Li et al. [18, 19] and Yu et al. [20] built up the models that could consider the melt-vapor-powder interaction, which greatly advances the modelling of coupling scheme in the context of LPBF. Nonetheless, the imperfections are the demanding computational resources and the lack of quantifications and a complete association between liquid, gas and powder. As such, it is significant to establish a unified model that integrates melt pool dynamics and powder motion considering the influence of metal vapor. Subsequent quantitative analysis stemming from such an integrated model would be highly beneficial.

In the present study, we endeavour to elucidate the dynamics of metal vapor flow and its consequent effects on powder flow during the Laser Powder Bed Fusion (LPBF) process through rigorous physics-based numerical simulation. Specifically, the developed model aims to provide both qualitative and quantitative perspectives on the generation of metal vapor and its subsequent evolution due to laser-material interactions. As will be subsequently demonstrated, the intricate interplay between the liquid, gas, and powder phases is comprehensively revealed.

2 Modelling

The modelling of powder motion is achieved by the discrete element method (DEM), which is widely used for modelling powder deposition/spreading process prior to the laser irradiation [15]. The computational fluid dynamics (CFD) method is employed to model melt flow in response to laser heat, adhering to the Navier–Stokes (NS) equations. Both the DEM and CFD approach have been successfully implemented in the separate research [16, 21, 22], to which can be referred. In addition to that, it is noticeable that the model should account for the evaporation of the liquid melt.

In the present modelling framework, the volume of fluid (VOF) model with high resolution interface capturing (HRIC) [23] method is used to track the metal-gas interface, where the vapor is treated as one of the constituents of the gas phase, mixed with the shielding gas, argon. Accordingly, to fulfil the mass conservation, the general form of the volume fraction equation should be segregated in terms of specific gas specie, expressed as [24]:

$$\left\{\begin{array}{c}\frac{\partial {\alpha }_{1}}{\partial t}+\nabla \cdot \left({\alpha }_{1}{\varvec{u}}\right)={m}_{LG}/{\rho }_{1}\\ \frac{\partial {\alpha }_{2}}{\partial t}+\nabla \cdot \left({\alpha }_{2}{\varvec{u}}\right)={-m}_{LG}/{\rho }_{2}\end{array}\right.$$

where \(t\), \({\varvec{u}}\), \({\alpha }_{1}\), \({\alpha }_{2}\), \({\rho }_{1}\), \({\rho }_{2}\) and \({m}_{LG}\) represent time, velocity, volume fraction of liquid and gas, density of liquid and gas, and mass transfer rate from the liquid to the vapor constituent. The mass transfer rate is calculated by [24, 25]:

$${m}_{LG}=-0.54{p}_{0}\mathrm{exp}\,({\mathrm{L}}_{\mathrm{v}}\mathrm{M}\frac{T-{\mathrm{T}}_{\mathrm{v}}}{RT{\mathrm{T}}_{\mathrm{v}}})\sqrt{\frac{\mathrm{M}}{2\pi RT}}\left|\nabla {\alpha }_{1}\right|$$

where \({p}_{0}\), \({\mathrm{L}}_{\mathrm{v}}\), \({\mathrm{M}}_{L}\), \(T\), \({\mathrm{T}}_{\mathrm{v}}\), and \(R\) represent ambient pressure, evaporation latent heat, molar mass of liquid, temperature, evaporation point, and gas constant, respectively.

As mentioned above, the gas phase comprises metal vapor and shielding gas, where the latter is initialized within the entire gas region. The mass fraction conservation equation is applied, which is also known as species transfer equation, can expressed as:

$$\frac{\partial }{\partial t}\left({{\rho }_{2}\alpha }_{2}{Y}_{i}\right)+\nabla \cdot \left({{\rho }_{2}\alpha }_{2}{\varvec{u}}{Y}_{i}\right)=\nabla \cdot \left({{\rho }_{2}\alpha }_{2}{D}_{i}{\nabla Y}_{i}\right)+{m}_{LG}$$

where \({Y}_{i}\) and \({D}_{i}\) represent mass fraction and diffusion coefficient for the two species, \(i=1, 2\) for the vapor and argon, respectively. All the properties related to the gas phase such as \({\rho }_{2}\) then convert to the mass fraction-weighted value.

2.1 DEM-CFD coupling

The co-existing of fluid phase (metal and gas) and DEM phase (powder) and their interaction are briefly introduced as follows. The volume fraction of fluid phase, \({\alpha }_{F\_i}\), is defined as the ratio of volume occupied by all fluid phases (\({\sum }_{i}{V}_{i}^{F}\)) to the total cell volume \(V\), and the volume fraction of DEM phase, \({\alpha }_{D\_j}\), is seen the rest volume fraction of the same cell. These are expressed as:

$$V={\sum }_{i}{V}_{i}^{F}+{\sum }_{j}{V}_{j}^{D}$$
$${\alpha }_{F\_i}=\frac{\sum_{i}{V}_{i}^{F}}{V}$$
$${\alpha }_{D\_j}=1-{\alpha }_{F\_i}$$

Based on the aforementioned coupling scheme, several modifications are necessitated in the computational framework. Specifically, the Navier–Stokes (NS) equation, Volume-of-Fluid (VOF) fraction equation, and species equation must be altered where the conventional volume fraction terms are substituted by \({\alpha }_{F\_i}\). Furthermore, the presence of fluid-powder interactions introduces an additional dimension; the source term in the NS equation and the energy equation, as denoted in [16], are augmented to incorporate the particle-induced force \({{\varvec{f}}}_{\mathrm{P}-\mathrm{F}}\) and energy \({q}_{\mathrm{P}-\mathrm{F}}\). For an exhaustive understanding of mass and heat transfer between the fluid and the DEM phase, readers are directed to the comprehensive introduction presented in [17]. The proposed model has been instantiated and tested using the Star-CCM + software platform [23].

2.2 Model simplifications

Considering the inherent intricacies associated with model formulation, several assumptions and simplifications have been proposed. These serve to alleviate challenges during the model construction phase, yet sufficiently retain the capacity to encapsulate the key phenomena under investigation. Besides, these can also supplement and clarify the ambiguity during the procedure of model construction that may be caused by unclear and verbose stacking of physical formulas.

  1. (1)

    Since powder phase is tracked by the DEM method, its deformation and transformation from solid to liquid is simplified by manually removing particles whose temperature exceed material liquidus line, which is named powder absorption effect in [26].

  2. (2)

    The formulation of laser absorption of powder phase is accomplished by a laser heat source term, which is proposed as:

$${q}_{p}=\frac{2{A}_{ap}P}{\pi {w}^{2}{D}_{p}}\mathrm{exp}(\frac{-2{r}^{2}}{{w}^{2}})$$

where \(P\), \(w\), \(\varepsilon\), and \(r\) denote laser power, laser effective radius, emissivity and the distance from laser centre. \({D}_{p}\) represents particle diameter. \({A}_{ap}\) represents laser incident angle (\(\theta\)) related absorption, expressed as [27]:

$${A}_{ap}=1-\frac{1}{2}\left[\frac{1+{(1-\varepsilon \mathrm{cos}\theta )}^{2}}{1+{(1+\mathrm{\varepsilon cos}\theta )}^{2}}+\frac{{(\varepsilon -\mathrm{cos}\theta )}^{2}}{{(\varepsilon +\mathrm{cos}\theta )}^{2}}\right]$$
  1. (3)

    Gas phase is treated as idea impressible gas and fluid flow is assumed to be laminar.

Taken the simplifications above, the constraints on spatial and temporal parameters are significantly relaxed. For mesh discretization, we employ adaptive meshing with a base grid size of 20 μm and the minimum grid size of 5 μm is used for the regions including the melt pool, metal-gas interface and metal vapor, and the 10 μm grid size is applied as transition. The time step of 1e-7 s is adopted for the simulations.

3 Simulation

This study adopts Ti-6Al-4 V as feedstock and substrate material whose thermos-physical properties are listed in Table 1.

Table 1 Temperature dependent thermo-properties of Ti-6Al-4 V and other miscellaneous parameters adopted in simulation, where T represents temperature (K)
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The proposed computational domain is shown in Fig. 1, where the vertical size of gas region and baseplate region are 1000 μm and 400 μm, respectively. A stationary laser beam is placed at the centre of X–Y plane. In addition, powder bed configuration is solely obtained by the DEM model, prior to initializing the DEM-CFD coupled model. The boundary surface for gas region and baseplate are pressure-based outlet and wall with temperature of 300 K, respectively.

Fig. 1
figure 1

Computational domain composed of 60 μm thickness powder bed, 400 μm cubic baseplate, and 1000 μm height of gas region (which is not shown here for refined visualization)

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4 Results and discussion

4.1 Identification of vapor flow states

Figure 2 presents a series of snapshots illustrating the gas velocity lines alongside the vapor profile (iso-surface) under a stationary laser. The vapor profile features four distinct iso-surfaces representing velocity magnitudes: 20 m/s, 50 m/s, 100 m/s, and 150 m/s, respectively. Each iso-surface is represented by a unique colour, as indicated by the magnitude scale. To enhance clarity in the visualization, the powder has been intentionally hidden from the display. Subsequently, the four distinct states of vapor flow under the stationary laser beam will be detailed.

  • (a) Initialization and continuation states:

    At the beginning scenario of t = 20 μs, vapor erupts rigorously from powder bed within a very short time. At the same time, the recoil pressure becomes operative, causing a depression in the melt pool surface. At this stage, the vapor velocity is relatively modest, with a peak value approximating 60 m/s. As this depression expands, the area of laser-material interaction grows, leading to the generation of a larger volume of vapor, which consistently ascends. The snapshots t = 50 μs and t = 70 μs (Fig. 2(a2/a2’) and a3/a3’) indicate that the rising vapor can reach the top computational boundary and then flow out. The horizontal width of the iso-surfaces of vapor velocity, which gradually expands from the points of surface depression up to the top boundary, suggests that the vapor might have achieved a stable flow state. The maximum velocity magnitude reaches over 150 m/s near the upper part of the depression. It is reasonable to infer that the vapor will maintain this flow situation until transitioning into the next phase.

  • (b) Transition state:

    The shape of vapor profile (velocity iso-surface), especially the 50 m/s iso-surface, begins to change at around t = 80 μs compared to t = 70 μs in the last state (continuation state). At approximately t = 90 μs (Fig. 2(b2/b2’)), vapor flows out of the right boundary. There is a noticeable vertical fluctuation in the width of the 50 m/s iso-surface, indicating the continuous flow is disturbed and vapor flow condition has already entered next state, termed here as “transition state”. The distinction of this state is that vapor flow direction is distorted, which becomes more evident at t = 100 μs (Fig. 2(b3/b3’).

  • (c) Interruption state:

    As melt surface depression deepens, the transition state ends at approximately t = 150 μs. This also indicates that the depression converts into the keyhole regime [30]. Note that the region where gas velocity is larger can also appears over the baseplate, as Fig. 2(c2) (t = 180 μs) show. Such an observation implies that the vapor flow has become discontinuous, exhibiting interruptions and oscillations. Such interruption of vapor may even cause transient dismission of the vapor within keyhole, and then restores from the keyhole bottom, shown as t = 250 μs (Fig. 2(c3/c3’)).

Fig. 2
figure 2

Different states of metal vapor flow, where with and without prime symbol (') represent velocity contour line and iso-surface, respectively. (a1)—(a3’) Vapor initially erupts and flows upwards continuously and straight. (b1)—(b3’) Vapor transition state where it is still continuous but with varied flow direction. (c1)—(c1’) Vapor interruption state, where vapor flow is unsteady and is periodically produced from keyhole bottom. (Note that when t = 20 μs, the scale for contour line of velocity magnitude is slightly different from others, as it is used to display mushroom-like vapor head, which is also a validation of the model. Similar mushroom-like shape was also observed in [31])

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In the keyhole regime, as depicted in the third panel of Fig. 2, a recurring pattern is observed: the peak vapor velocity near the top of the keyhole correlates with periodic shape changes of the keyhole wall. This phenomenon is further illustrated in Fig. 3. Since laser is vertically impacting on the keyhole wall, its absorption will enhance when the regional angle (\({\uptheta }_{1}\) and \({\uptheta }_{3}\)) between the keyhole wall and radial direction narrows [32]. In contrast, laser heat is minimally absorbed when this angle (\({\uptheta }_{2}\)) is larger than 90 degrees. In the former circumstance, metal vapor is more easily to occur under smaller angle (\({\uptheta }_{1}<{\uptheta }_{3}\)), resulting in elevated vapor velocity, which occasionally may exceed 500 m/s [24]. Normally, this fluctuation is known to be induced by the competitive relationship between recoil pressure and Marangoni force [33], which collectively drive the dynamic flow of melt along the keyhole wall. In the scope of current study, evaporation is also pinpointed as a factor that contributes to altering the regional keyhole wall angle. This alteration is brought about by the loss of liquid mass, consequently generating a zone of minimal absorption, as shown in Fig. 3 (No absorption region). As a result, the combined effect of Marangoni force, recoil pressure, and evaporation jointly modulate laser absorptivity by altering laser incident angle in specific regions of the keyhole wall, which finally leads to the interruption state of vapor flow.

Fig. 3
figure 3

Keyhole wall condition that results in the interruption state of vapor flow

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4.2 Verification of vapor flow

The above illustrated vapor evolution aligns impressively with the experimental observations made by Bitharas et al. [5]. Figure 4 displays the comparison in terms of vapor flow states between numerical simulation and experiments. For the purpose of generalization, four different snapshots from that used in Fig. 2 are exhibited, including t = 30 μs, 60 μs, 140 μs and 380 μs, respectively. In the experimental setup, capturing the nuances of vapor flow below a 1 mm height from the substrate is challenging due to the limited resolution of schlieren photography. Consequently, the depiction of vapor plume relies primarily on the elevated position [4, 5]. However, the current simulation clearly reveals that the chaotic plume observed in Fig. 4(d) is attributed to the interruption of continuous vapor flow combined with the varied flow direction, which is further exacerbated by the periodical vapor emission at the keyhole upper.

Fig. 4
figure 4

Comparison in vapor flow states between experiments [5] and simulated results. a/(a’) Initialization state, (b)/(b’) continuation state, (c)/(c’) transition state and (d)/(d’) interruption state (Note that the length and time scale are not exactly consistent, which has been discussed in the corresponding context, and the comparison here is mainly qualitative)

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To provide a quantitative assessment of the different vapor flow states, the vertical and horizontal vapor velocity magnitude on the central longitudinal section is extracted across four states. Specifically, t = 30 μs, 60 μs, 140 μs, and 250 μs are chosen for analysis, respectively, as Fig. 5 shows. The velocity value in each cell is obtained and marked, yielding three salient observations:

  • (1) Vertical velocity distribution:

    Across all states, the vertical velocity peaks near the upper part of the depression or keyhole (near the baseplate positioned at Z = 0.4 mm). In the initialization state, velocity diminishes at Z = 1.2 mm given vapor does not reach top boundary (at Z = 1.4 mm). In both the continuation and transition states, vapor maintains a continuous flow out of the top boundary. Meanwhile, during the interruption state, there's a velocity drop near the baseplate surface (Z = 0.4 mm) followed by a rapid acceleration.

  • (2) Horizontal velocity distribution:

    For the initialization and continuation states, the distribution of horizontal velocity is symmetrical. This symmetrical distribution starts to show disturbances during the transition state and is entirely disrupted in the interruption state.

  • (3) Velocity tends to peak at the transition state:

    The transition state registers the highest velocities. As the laser delves deeper into the baseplate, the surface area of the keyhole wall expands. Consequently, evaporation occurs across virtually the entire keyhole wall. By contrast, during the interruption state, evaporation is primarily localized at the upper region of the keyhole.

Fig. 5
figure 5

Vertical (first row) and horizontal (second row) velocity magnitude in each cell on the central plane of each vapor states: (a)/(a’) Initialization state at t = 30 μs, (b)/(b’) Continuation state at t = 60 μs, (c)/(c’) Transition state at t = 140 μs, and (d)/(d’) Interruption state at t = 250 μs

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4.3 Powder flow mechanism

Due to the vigorous flow condition of the metal vapor, powders are compelled to displace from their original positions. To provide a coherent and comprehensive visualization, the same time frames as discussed in Fig. 2 are used. Here, the powder flow snapshots are presented from an isometric viewpoint. Specifically, the initial three states are depicted in Fig. 6, while the interruption state is showcased in Fig. 7. For a more dynamic and holistic understanding of the powder flow evolution, we've included a video in the Supplementary materials.

  • (1) From t = 20 μs to t = 100 μs:

    Figure 6 exhibits the snapshots of the first three states of vapor evolution including initialization, continuation and transition. Rather than large-scale displacement, powders mainly undergo melting (or removed in the current model) under the influence of high-energy density laser irradiation. Still, powder in the vicinity of laser spot tends to flow toward melt pool and then showcases potential upward trajectory. The rest of powder remains largely unaffected. The subdued powder velocity can be attributed to the modest average vapor velocity magnitude, which is not powerful enough to displace powders significantly from the baseplate. The powder streamlines in Fig. 6(c1) and (c2) also indicate that metal powder mainly follows two flow patterns: an initial movement toward the melt pool followed by an upward lift. As powder approaches melt pool region, its temperature increases fast, and powder emerges into the melt pool. This is also the main cause of denudation phenomenon [34], which may decrease the number of powder melted during the scanning of adjacent tracks. Nevertheless, few powder can overcome the combined constrain effect exerted by the melt pool and low vapor velocity, producing the maximum velocity of powder is no more than 5 m/s.

  • (2) From t = 150 μs to t = 250 μs:

    As displayed in Fig. 7(a1) - (a3), more powder particles are propelled far away from the baseplate, especially at t = 250 μs. Beyond the upward movement, powder also exhibits outward flow pattern. The corresponding powder streamlines, as depicted in Fig. 7(a1’) - (a3’), indicate a shift in the mainstream direction of powder flow. This change largely stems from the vapor's altered flow direction starting from the transition states. Furthermore, the periodic surges in vapor velocity imbue the powder with substantial acceleration potential. As vapor expands, the powder is impelled outwardly at augmented velocities. Nevertheless, the peak velocity of the powder does not surpass 10 m/s.

Fig. 6
figure 6

(a1)—(a3) Correspond to vapor initial stage where vapor just spouts out and keeps continuous flow. (b1)—(b3) indicate transition state. (c1) and (c2) represent corresponding powder streamlines of (a3) and (b3), respectively

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Fig. 7
figure 7

(a1)—(a3) Powder spatter and (a1’)—(a3’) Corresponding powder streamline at 150 μs, 180 μs, and 250 μs, respectively, observed from specified view

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4.4 Implications for real-world LPBF

While the present study primarily concentrates on the stationary laser, its findings can also provide valuable insights for the real-world LPBF manufacturing. First, the present model explicitly unravels the vapor flow conditions in the presence of the vapor depression or a keyhole. The stages of vapor changes with the evolution of the depression or keyhole. This situation gradually parallels the cases when the laser scan speed decreases. Second, examining stationary laser serves as a foundational understanding of the liquid–gas-powder interaction, which acts as a baseline or a stepping stone, from which we can incrementally introduce the intricacies associated with the moving laser. Laser but not least, Bitharas’ experiments [5] have identified similarities in the behavior of metal vapor and other involved phases between the stationary laser and moving laser. Such empirical observations further facilitates investigations into the stationary laser, suggesting that the derived insights would be transferable, at least in part, to the more complex moving laser scenarios.

5 Conclusion

Utilizing the coupled DEM-CFD model, this study unravels the intricate relationship between the flow state of metal vapor, resulting from laser-material interactions, and the associated powder spatter patterns. Key conclusions drawn are:

  1. 1.

    The progression of vapor flow condition can be categorized to four distinct states: initialization, continuation, transition, and interruption, respectively. The initial two states witness a continuous and vertically directed vapor flow with a symmetric horizontal velocity distribution. However, by the transition phase, this vapor flow direction begins to diversify. The interruption state conforms to the keyhole regime, where keyhole wall fluctuates periodically, influenced by the interplay of the recoil pressure, Marangoni force, and mass loss, induce periodic vapor emergence at the upper keyhole.

  2. 2.

    Corresponding to the vapor flow states, the powder flow exhibits three different patterns. Initially, the powder is dragged towards the melt pool at minimal velocities during the first three vapor states. Subsequently, the escalating vapor velocity elevates the powder. In the final phase, the expansive nature of the vapor propels the powder away from the baseplate, causing it to flow outward.