Experimental and numerical investigation of the origin of surface roughness in laser powder bed fused overhang regions

ABSTRACT Surface roughness of laser powder bed fusion (L-PBF) printed overhang regions is a major contributor to deteriorated shape accuracy/surface quality. This study investigates the mechanisms behind the evolution of surface roughness (Ra) in overhang regions. The evolution of surface morphology is the result of a combination of border track contour, powder adhesion, warp deformation, and dross formation, which is strongly related to the overhang angle (θ). When 0° ≤ θ ≤ 15°, the overhang angle does not affect Ra significantly since only a small area of the melt pool boundaries contacts the powder bed resulting in slight powder adhesion. When 15° < θ ≤ 50°, powder adhesion is enhanced by the melt pool sinking and the increased contact area between the melt pool boundary and powder bed. When θ > 50°, large waviness of the overhang contour, adhesion of powder clusters, severe warp deformation and dross formation increase Ra sharply.


Introduction
Laser powder bed fusion (L-PBF) is an advanced additive manufacturing (AM) technique wherein metal powder is selectively fused using a focused laser beam to build up a three dimensional (3D) metallic part layer by layer according to sliced 3D computer-aided design (CAD) models (Chatham, Long, and Williams 2019;Tan, Zhu, and Zhou 2020). Depending upon whether material exists below the printed layer or not, the printed region can be classified as either a solid region or an overhang region, respectively. An overhang region is thus a special structure that is built not on a solid substrate but rather directly on top of the powder bed (Patterson, Messimer, and Farrington 2017). Overhang regions can be built with or without support structures, and the L-PBF of overhang regions with supports is similar to the process on a solid substrate (Wang and Chou 2018) with the exception that the supports are built with lower density (and hence lower mechanical strength) than solid substrates, making it easier to mechanically remove them after the L-PBF process. Overhang regions printed with support structures thus need additional post-processing steps, namely support removal, grinding, and polishing after the L-PBF process.
In some particular cases such as the fabrication of horizontal interior channels, the overhang region in the top half of the channels has to be built without supports because of the difficulties of removing the supports after the process (Hopkinson and Dickens 2000). In addition to support structures not being possible for horizontal internal channels, it is also difficult to implement any surface finishing processes in the internal surfaces, especially for complex 3D channel networks encountered in conformal cooling channels (Feng, Kamat, and Pei 2021). As a result, the overhang region can deviate from its designed shape owing to: (i) residual stress-induced deformation, (ii) staircase effect (Kuo et al. 2020;Li et al. 2020), and (iii) enhanced surface roughness caused by undesired powder sintering; where, the former two factors are classified into 'macro' deviations usually in the mm length scale and the latter one is recognised as 'micro' deviation usually in the µm length scale.
Thermal stress-induced deformation is an important problem encountered in overhang regions (Patterson, Messimer, and Farrington 2017). The local melting/ cooling induces large temperature gradients in and around the melt pool causing intensive thermal stresses in the solidified layers. Thermal stress-induced warping does not deform the solid regions appreciably since these regions are constrained by many layers below; on the other hand, the overhang regions are unconstrained and suffer significant deflection due to stress relief during the process (Kamat and Pei 2019). Moreover, the melt depth is larger than the layer thickness (since previous layers are also remelted to ensure sufficient bonding between the built layers [Yadroitsev et al. 2013;Kamath et al. 2014]), which induces a shape deviation (such as dross [Charles et al. 2020;Feng et al. 2020]) since the solidified thickness is larger than the designed one. In the microscale, printed surfaces (R a and S a ∼ 10 μm) are rougher than mechanically machined surfaces (Duval-Chaneac et al. 2018;Wen et al. 2018). This problem is especially severe for overhang regions whose surface roughness (R a ) is usually around 20 μm as a result of unmelted powder adhering to the edges of the solidified melt pool (Mazur et al. 2016;Pakkanen et al. 2016).
The overhang angle (θ, measured with respect to the build direction) is a crucial parameter that affects the warp deflection of overhang regions as well as the surface roughness (Kamat and Pei 2019;Mingear et al. 2019). An overhang angle of θ ∼ 45°is generally agreed upon as the critical value up to which overhang regions can be printed without support structures (Pakkanen et al. 2016;Kadirgama et al. 2018). When θ is larger than this critical value, the overhang regions cannot be printed with acceptable surface quality. Besides the overhang angle, laser parameters (related to laser energy density) also influence the surface roughness of the overhang regions by affecting the shape/size of the melt pool as well as the melt pool dynamics (Wang et al. 2013;Mingear et al. 2019).
Melt pool dynamics is a common physical phenomenon in laser materials processing, including the L-PBF processes, conducted on both the solid (Shrestha and Chou 2018) and overhang (Le et al. 2020) regions. The melt pool shape, size, and cooling rates affect both the residual stress-induced deformation as well as the surface roughness, thus acting as a bridge between processing parameters and surface morphology/quality and meriting further investigation using numerical simulations to understand the melt pool behaviour and its effect on surface roughness. To date, several studies have been conducted to simulate the melt pool behaviour during L-PBF of solid regions. Simulation techniques such as finite element method (FEM) (Roberts et al. 2009;Du et al. 2019), finite difference method (FDM) (Wu et al. 2018), computational fluid dynamics (CFD) (Lee and Zhang 2016), and arbitrary Lagrangian-Eulerian method (ALE) (Khairallah and Anderson 2014) were employed to study the heat transfer (temperature field) and mass transfer (melt flow) processes with the inclusion of evaporation recoil pressure  and Marangoni convection ) phenomena. Further, the discrete element method (DEM) was used to create a random-distributed powder bed (Lee and Zhang 2016;Wu et al. 2018). These models simulated the L-PBF processes from the powder-scale (Khairallah et al. 2016) to mesoscale (Khairallah and Anderson 2014), and from single-track (Leitz et al. 2017) to multitrack (Foroozmehr et al. 2016) and multi-layer (Huang, Khamesee, and Toyserkani 2019).
However, the melt pool dynamics in overhang regions which dictates the resulting surface roughness has received scant attention in the literature. Although existing simulation models for L-PBF of solid regions can serve as a reference to some extent, there are considerable differences in the melt pool dynamics between the overhang regions and solid regions. In overhang regions, the molten metal flows downwards into the gaps between powder particles, making the melt pool sink under the weak support offered by the porous powder bed. This makes the effects of gravity and surface tension important in the determination of the resulting shape/size of the melt pool and, subsequently, the evolution of the microscale morphology of overhang regions. Moreover, due to the voids between powder particles, thermal conditions (such as energy absorption, energy loss, cooling rate, and thermal conductivity) of printing on a powder bed are inferior to printing on a solid substrate, worsening the surface quality, porosity, and metallurgical strength of the resulting part Karimi et al. 2020;Sing and Yeong 2020). Surface roughness not only increases the (micro-) shape deviation but also deteriorates the mechanical strength by serving as initiation sites for micro-cracks during periodic loading (Günther et al. 2018). The high surface roughness of overhang regions limits the application of L-PBF in the fabrication of parts that have strict requirements on (micro-) accuracy/quality. The present study systematically and comprehensively investigates the formation mechanisms of micro shape deviation of overhang regions (built without supports) and the origins of surface roughness using experimental and simulation studies. A coupled DEM-CFD simulation model is developed to reveal the melt pool dynamics of the overhang region and formation mechanisms of surface morphology, taking into account the effects of border track contour, powder adhesion and warp deformation. The surface roughness R a is studied as a function of overhang angle using simulations and single-factor L-PBF printing experiments. Three mechanisms of powder adhesion in the overhang regions associated with the sinking of the melt pool are identified and discussed in detail. Finally, potential solutions to mitigate high surface roughness issue in as-printed overhang regions are discussed briefly.

Experimental setup and materials
The SLM 125HL machine (SLM Solutions, Germany) was used to conduct the L-PBF experiments. A random-polarized Gaussian beam (M 2 ∼ 1.19) was produced by a fibre laser with a wavelength of 1070 nm and a focused beam diameter of 70 µm. The processing was conducted in a protective argon (Ar) atmosphere (overpressure 17 mbar). Spherical powder of 17-4 PH stainless steel (LPW Technology, UK) with a size distribution in the range of 10-45 μm was used as the experimental material in the present study. The manufacturer-specified chemical composition of the alloy is listed in Table 1.

Printing of overhang region with varying overhang angles
The shape and size of the L-PBF printed samples are illustrated in Figure 1. The samples had an overhang region with θ = 0°, 15°, 30°, 45°, 50°, 55°, and 60°, respectively. The overhang regions for all the samples had the same length of 12 mm (except for the 0°cubic sample which had a 12 mm-tall vertical wall). Attempts to print overhang regions with θ greater than 60°were also made but failed during printing due to large tip warpage in the overhang region. The process parameters for the solid region (both interior and border) and overhang region (border only) were selected, as listed in Table 2, according to the recommendation of the manufacturer, which were optimised to minimise porosity in the printed samples. The solidified layer thickness was 30 μm and the build plate was preheated to 100°C during the L-PBF process.
Each layer was double contoured (i.e. an inner border and an outer border with an 80 μm spacing between the inner and outer borders). The borders in an overhang region are classified into three situations depending on the overhang angle: two solid borders (θ ≤ 30°), one solid border (inner) and one overhang border (outer) (30°< θ ≤ 50°), and two overhang borders (θ > 50°), as shown in Figure 2. The printing sequence was as follows: outer border of the solid region, inner border of the solid region, the inner border of the overhang region (if applicable), outer border of the overhang region (if applicable), and finally the interior solid region.
The surface micro-morphology as well as the surface roughness in the overhang region was measured using an Alicona optical 3D measurement system (InfiniteFocus G5 plus) wherein the vertical resolution was 0.1 µm under 10× magnification. The measurement of surface roughness was conducted using the methodology of ISO 4288 ("Geometrical Product Specifications (GPS) -Surface Texture: Profile Method -Rules and Procedures for the Assessment of Surface Texture (ISO 4288)" 1996). In this standard, the roughness sampling length is recommended as 8 mm when R a > 10 µm. Three points on each sample (Locations A, B, and C in Figure 1, corresponding to three distances from the start point of the overhang: 5, 6, and 7 mm, respectively) were selected to measure the surface roughness. Linear measurement was performed along two orthogonal directions (Directions I and II in Figure 1) at each measuring location. Cross-sectional micrographs are important in observing the microstructures and roughness profiles (Cai et al. 2019(Cai et al. , 2021. Micrographs of the cross-sectional overhang regions were obtained with optical microscopy to illustrate the mechanisms of powder adhesion. The samples were mounted into a resin embedding, ground, and polished to the final step with 1 µm diamond before the optical microscopy.

Single-track printing
To verify the fidelity of the simulation model, singletrack printing was conducted on a 17-4 PH stainless steel substrate. The experiments were conducted under the same process parameters as the printing of interior solid region (listed in Table 2) except for hatch spacing and scanning strategies that are not applicable for single-track printing. The transverse cross-section of the printed sample was ground, polished, and etched to reveal the melt pool boundary. The profile of the track contour was then observed and measured using a scanning electron microscope (SEM). The results will be presented in Section 3, where experimental and simulation-predicted melt pool dimensions will be compared.

Single-factor experiments for laser power and scan speed
To find out the potential solutions of high surface roughness issue in L-PBF printed overhang regions, the effects of laser power and scan speed on the surface roughness of the overhang region were studied using single-factor experiments. Samples with 45°overhang angle were built under five levels of laser power (70, 85, 100, 115, and 130 W) and five levels of scan speed (700, 850, 1000, 1150 and 1300 mm/s), where the power and scan speed refer to the overhang border parameters. The solid parameters and other overhang border parameters (except for laser power and scan speed) were maintained constant according to the manufacturer-recommended values listed in Table 2.

Simulation details
A coupled DEM-CFD model was developed to simulate the melt pool behaviour and predict the shape/size of the melt pool and solidified track contour. A powder bed with randomly generated particle sizes (within the range of 10-45 μm) and locations was created using   Figure 2. Borders in the overhang region depending on the overhang angle θ: (a) two solid borders when θ ≤ 30°, (b) one solid border and one overhang border when 30°< θ ≤ 50°and (c) two overhang borders when θ > 50°.
Yade (Šmilauer et al. 2015), an open-source DEM software. Due to the voids between powder particles, the apparent density of the powder bed was much lower than the density of a bulk solid. This resulted in powder bed shrinkage during the printing process, causing the thickness of the solidified layer to be smaller than the thickness of the original powder bed (Li et al. 2018;Tan et al. 2019). The solidified layer thickness was around 0.5 times the thickness of the powder bed layer according to our simulation results and the experimental results in the literature (Yadroitsev et al. 2013). Therefore, to obtain a 30 μm-thick solidified layer (which remains consistent with the experimental value), the thickness of the powder bed layer was selected as 60 μm. To import the geometry model of powder bed into the CFD model, the meshing data was saved into a binary STL file. The L-PBF process was modelled using a commercial CFD software (FLOW-3D, USA). The modelling space was meshed by cubic cells having a size of 4 × 4 × 4 µm. To save computational time and cost, only a small geometry where the track was printed was modelled, not the whole substrate and powder layer. For the purpose of modelling, the model geometry was constrained by 'virtual' boundaries. Heat transfer at the boundaries was continuous and smooth, and the boundary conditions were therefore set as semi-infinite . The edges of the melt pool were set as free-slip wall boundaries, where the normal component of the fluid velocity field was equal to zero while the tangential component was unrestricted. The molten metal was assumed to be an incompressible fluid to which a static pressure condition was applied. The physical phenomena occurring during the L-PBF process, such as heat transfer, vaporisation, constant pressure bubbles, air entrainment, temperature-dependent surface tension (driving Marangoni convection and capillary effect in the melt pool), viscous flow and gravity, and solidification with shrinkage and microporosity, were considered in the simulation. The momentum and continuity equations were solved numerically (Flow-3D V11.2 Documentation 2016): Mass continuity equation Momentum equations where t is time; ρ is the molten metal density; (u, v, w) are velocity components in the coordinate directions (x, y, z); (A x , A y , A z ) are the fractional area open to flow in the (x, y, z) directions; R SOR is a mass source; V F is the fractional volume open to flow; p is the pressure; (G x , G y , G z ) are body accelerations, and ( f x , f y , f z ) are viscous accelerations. Both explicit solvers and implicit solvers were employed to solve the equations wherein explicit solvers were used for the physics of free surface pressure, viscous stress, and advection (first-order) and implicit solvers were used for the physics of heat transfer and surface tension pressure. The dynamic time step (between 10 −8 -10 −7 s) was automatically chosen during the simulation. An HP Z6 G4 workstation configured with a 28-core CPU (2.00 GHz frequency) and 64 GB DDR3 memory was used to perform the simulations. The material properties used in the simulations are listed in Table 3.
In laser materials processing, the laser beam irradiates the substrate and penetrates it till a certain depth. For a Gaussian laser beam, the 3D volumetric laser energy source can be conveniently approximated to be in the shape resembling an circular paraboloid inside the substrate (Figure 3a), where the base diameter is equal to the laser beam spot diameter (70 µm in our case) and the height (representing the optical penetration depth of laser beam) is related to the energy attenuation inside the powder bed. The 3D shape of the volumetric heat source is determined by: where r, r 0 , z and z PD are the radial coordinate, laser spot radius, depth coordinate, and the optical penetration depth, respectively. The rationale behind the functional form of the volumetric heat source used in Eq. 5 is discussed in Appendix A. The optical penetration depth (i.e. the height of the volumetric heat source) was determined through a series of trial simulations to ensure that model predictions agreed well with the experimentally measured melt pool dimensions. More specifically, simulations of single-track printing on the solid substrate were performed using different optical penetration depths, i.e. setting different heights for the volumetric heat source. The optical penetration depth at which the simulated melt depth was closest to the experimental melt depth was selected as the calibrated optical penetration depth and used in subsequent simulations. In these single-track printing simulations, the process parameters were the same as stated in Section 2.2.2. To simplify the model and ensure easy implementation in the Flow-3D simulation software, the laser power (200 W for the interior solid region) was assumed to be uniformly distributed within the circular paraboloid volume. The model geometry is illustrated in Figure 3b and the comparison of melt pool dimensions obtained from the experiment and simulation is shown in Figure 3c. The calibrated optical penetration depth value of the equivalent volumetric heat source was found to be 110 µm which is physically reasonable since similar melt depths have been measured before in the literature under similar process parameters (Kamath et al. 2014). Figure 3c shows that the calibrated model was capable of predicting the shape and the dimensions of the resulting melt pool accurately. This calibrated model was used to simulate the L-PBF process of the border tracks in the overhang region. The model geometry is illustrated in Figure 4. The process parameters used in the simulation are listed in Table 2. To save the computational time and cost, the spherical-shaped powder was converted into 32-facet polyhedrons to reduce the number of mesh nodes.

Melt pool dynamics in the overhang region
The DEM-CFD model described in Section 3 was used to gain physical insight into the formation of surface roughness in overhang regions. The solidified contour of tracks in a 45°overhang region is illustrated in Figure 5. For a 45°o verhang region, a solid inner border was printed first followed by an overhang outer border, corresponding to the tracks 1 and 2 in Figure 5, respectively. Tracks 1 and 2 were simulated using the process parameters listed in Table 2. Here, the solidified contour of track 2 was the result of the evolution process shown in Figure 6. Figure 6 illustrates the evolution of the melt pool in the overhang region (outer border) with a 45°overhang angle. When the laser beam moved towards and then away (perpendicular to the plane) from the area of interest (highlighted in Figure 6 with a dashed rectangle), the powder was first heated by the laser beam and then cooled down due to heat diffusion to surrounding material and the convective Ar atmosphere. Correspondingly, the evolution process of a melt pool can be roughly divided into two stages: melting (including powder melting and gap filling) and solidification. There is a transient duration (875-879 μs) from the melting stage to the solidification stage with no clear demarcation between these two stages. During the melting stage, the powder began to melt forming a melt pool until the melt pool enlarged to the maximum depth and then maximum width. The top of the powder bed (powder particle A in Figure 6) was melted first as it was directly exposed to the laser irradiation. The width and depth of melt pool increased over time as the laser energy was transferred to the surrounding powder mainly via heat conduction. Gaps between powder particles were subsequently filled up by molten metal via the gravity and the capillary effect. Powder at the centre of the melt pool (particle A in Figure 6) was completely melted while powder particles at the border of the melt pool (particles B, C, and D in Figure 6) were partially melted.
In the solid region, the melt pool exists on a dense substrate. The depth of the melt pool in this case is mainly determined by the laser energy transmission and heat transfer inside the substrate but not the gravity or surface tension. On the other hand, both gravity and surface tension have considerable effects on the depth of the melt pool in the overhang region where the melt pool is formed on a powder bed containing many voids. As a result, molten metal flows downwards into the voids inducing the sinking of the melt pool, increasing the depth of melt pool and promoting dross formation. The sinking of the melt pool is thus driven by gravity and surface tension-induced capillary effect. In Figure 6, gaps 2 and 3 were filled via this phenomenon. Melt pool sinking was aided by the temperature-dependent viscosity since high temperatures improved the flowability of the molten metal and thus sunk the melt pool further, and vice versa. At a time of 875 μs, when  the melt pool reached its maximum depth, the upper half of gap 4 was surrounded by the molten metal. However, molten metal was unable to penetrate this gap owing to the limiting effect of viscosity. The molten metal subsequently cooled down and solidified, leaving this gap unfilled as a pore in the overhang region (Figure 10e). It is interesting to note that the melt pool reached its maximum depth (875 μs) and width (879 μs) asynchronously, although the time difference is quite small. Not all the gaps in the melt pool were filled when the melt pool reached its maximum depth. There was a 3 μs lag between maximum melt depth and all gaps being filled. Moreover, the volume of the fully melted region (red region in Figure 6 that the liquid fraction equals 1) reached its maximum at a time of 885 µs, 10 μs later than the maximisation of melt depth and 6 μs later than the maximisation of melt width. During the solidification stage, the melt pool cooled down and solidified gradually via thermal diffusion into the solidified layers and forced convection of Ar flow. The cooling rate was much lower than the heating rate of the laser due to the extremely high laser power density. Thus, the duration of the solidification stage was around 6 times longer than that of the melting stage. The cross-sectional shape of the solidified melt pool manifested as the track contour.

The formation mechanism of surface morphology in the overhang region
The surface morphology of an overhang region is mainly determined by a combination of the solidified contour of the border tracks, powder adhesion, and thermal stresses-induced warp deformation, as explained in the subsections below.

Effect of the solidified border contour
Ideally, the overhang region is a flat surface (looking like a straight line in the Y-Z cross-section, as shown in Figure  5), but in reality, border track contours of multilayers manifest themselves as a wavy contour of the overhang region. Powder adhesion due to partial sintering further imparts irregularity to the morphology of this wavy contour, increasing its roughness as discussed in Section 4.2.2. Because the overhang region is an assembly of sliced layers, when not considering powder adhesion, the contour of the overhang region can, in principle, be deduced from the regularly duplicated array of a single-layer contour following the specific overhang angle and layer thickness, as shown in Figure  7. When θ ≤ 60°, the contour of inner borders is overwritten by the outer borders ( Figure 7a). Therefore, the outer borders are the main contributor to the overhang contour. When θ > 60°, both the inner and outer borders contribute to the overhang contour ( Figure  7b). The resulting wavy contour of the overhang region is a source of surface roughness.

Effect of powder adhesion
As discussed in Section 4.1, powder on the edges of the melt pool can be partially melted or sintered, e.g. powder particles B, C, and D in Figures 5 and 6. The remaining unmelted part is exposed to the overhang region (known as powder adhesion), contributing to the surface morphology. When considering powder adhesion, as shown in Figure 8, the contour of the overhang region becomes irregular, thus enhancing the surface roughness. Although powder adhesion is stochastic in nature, both the experimental and simulation results indicated that there are statistical correlations between the process parameters (overhang angle, laser power, and scan speed) and the surface roughness of the overhang region. Powder adhesion is the main contributor to surface roughness in the overhang region. In addition to powder adhesion, the solidified border contour and warp deformation jointly enhance surface roughness when θ > 45°. Further discussion will be presented in Section 4.3.1.

Effect of warp deformation of the overhang region
When printing an overhang region without supports, thermal stresses induce warp deformation in the overhang region due to the lack of constraints. Although warp deformation is a measure of macro shape deviations, it has a combined effect with the other two influencing factors (border track contour and powder adhesion) on the surface roughness of the overhang region. As discussed in Section 4.2.1, the contour of the border tracks determines the morphology of an overhang region. The contour of the overhang region is determined by the relative position relationship between the border tracks of adjacent layers. When θ ≤ 45°, the effect of warp deformation is negligible, as shown in Figure 9a, b and c. The offset of the border contour of layer i + 1 with respect to the border contours of layer i is calculated by where Δ V , Δ H , and t are vertical offset, horizontal offset, and layer thickness, respectively. When θ > 45°, the effect of warp deformation on the contour offset has to be taken into account, as shown in Figure 9d and e. The warp deformation mainly S74 induces shape deviations in the vertical direction while its effect in the horizontal direction is negligible. The vertical offset of layer i + 1 with respect to layer i was estimated by Kamat et al. (Kamat and Pei 2019) to be: where σ y , m, n, j, and E are yield stress, melt depth-tolayer thickness ratio, number of total layers, index of summation (i + 1 ≤ j ≤ n), and Young's modulus, respectively. It is concluded from Eq. 8 that the warp deformation-induced contour offset becomes more considerable with the increase of the overhang angle.

Effect of overhang angle on surface roughness in the overhang region
4.3.1. Estimation of the surface roughness in the overhang region from simulation As discussed in Section 4.2, the surface roughness in the overhang region is a result of the combined effect of border track contour, powder adhesion and warp deformation. In light of this conclusion, the 2D multi-layer contour of the L-PBF printed overhang region can be approximated using the simulation results of border tracks by taking into account the powder adhesion and warp deformation-induced contour offset (Eqs. 6-8). Specifically, seven cross-sections were selected randomly from the overhang region and then arrayed according to the calculated offsets in horizontal and vertical directions (as shown in Figure 10a and b). Profiles of the border tracks including the adhered powder particles were extracted as the predicted multi-layer overhang contour, as shown in Figure 11. This approximation is a more efficient and economical way to simulate the micro surface deviation of overhang regions and estimate the surface roughness as 3D multi-layer simulations are much more complicated and cost-intensive. The simulation-predicted 2D contours of the overhang regions with various overhang angles are extracted and plotted in Figure 11. The dashed lines indicate the planar profile of designed overhang region specified by the overhang angles. R a was calculated by averaging the contour offset from the reference line:   where x is the coordinate parallel to the reference line, δ is the contour offset from the reference line and is a function of x, and L is the sampling length (varying from 210 µm to 1209 µm depending upon the overhang angle). It is seen that the powder particles adhered to the overhang region forming surface roughness. The contour was in good accordance with the reference line when θ ≤ 45°, while a deviation between the contour and the reference line became noticeable when θ > 45°. This deviation was due to the warp deformation discussed in Section 4.2.3. For a sampling length of 1190 µm when θ = 80°, according to Eq. 8, the deviation in the vertical direction was as large as 123 µm. Thus, the contour deviated largely from the reference line at θ = 80°, as shown in Figure 11e.
The experimentally measured R a in overhang regions is shown in Figure 12a as a function of varying overhang angle. Overall, R a increased with increasing overhang angle. Moreover, it was found that the effect of measuring direction was less significant, indicating that R a was isotropic and independent of direction. A comparison of simulation-predicted R a and experimentally measured R a is shown in Figure 12b. Here, both the experimental and simulation-predicted roughness values were normalised to the reference value of R a at 0°overhang angle. It should be noted that the absolute value of the simulation-predicted R a is smaller than the experimental results (around half of the experimental result when θ = 0°) because the sampling length in the simulation is much smaller than the specified value in ISO 4288, making some deviation information unavailable. In general, the simulation-predicted trend of R a was consistent with the experimental results although there was a discrepancy when θ ≥ 60°due to the considerable but unpredictable dross in the overhang region.
The overhang angle is the most important structural parameter for an overhang region, indicating the degree of inclination of an overhang region and determining the difficulty of L-PBF. The overhang angle plays a decisive role in the surface morphology and shape deviations on both the macroscale and microscale, influencing stress-induced deformation and surface roughness, respectively. On the macroscale, as shown in Figures 9 and 11, the L-PBF printed samples displayed good shape accuracy when θ ≤ 45°, while shape deviations became a considerable problem visible to the naked eye when θ > 45°. The shape deviation was mainly caused by the upward warp at the overhang tip. This was especially significant when the overhang angle was as large as 60°where severe warpage and dross were observed at the tip. On the microscale, meanwhile, surface morphology became rougher gradually with increasing overhang angle, indicating an increase of surface roughness on the overhang region, as shown in Figures 11 and 12a.
From the simulation and experimental results, the changing trend of R a with the overhang angle can be roughly divided into three regimes (Figures 11 and  12a). First, there was no change of R a with the increasing overhang angle until θ = 15°. Then, R a gradually increased with the increase of overhang angle when 15°< θ < 50°. The rate of change of R a with respect to overhang angle increased sharply at θ = 50°, after which the R a curve increased rapidly. The relation between the surface roughness of the overhang region and the overhang angle was mainly established by the mechanisms of powder adhesion, where powder or powder clusters protruded from the surface (Figure 13).
The mechanisms of powder adhesion were strongly related to the overhang angle. In other words, different ranges of overhang angle displayed different mechanisms of powder adhesion. R a at θ = 0°was considered the benchmark for comparison. Here, there was no overhang region, and the melt pool was fully supported by the previously printed layer (substrate). Only a small area of the melt pool boundaries directly contacted the powder bed at the side which resulted in slight powder adhesion on the side walls. When θ ≤ 15°, the overhang angle was small enough and thus its effect on R a was negligible (Figures 11a and 13a).
When the overhang angle ranged from 15°to 50°, part of the melt pool was supported by the powder bed. This implied: (a) the weakness of the support of the melt pool, promoting the sinking of the melt pool and inducing more molten metal flowed downwards into the gaps between powder particles; (b) the increase of contact area between the melt pool boundary and powder bed. Both of these two factors resulted in more powder particles being adhered, promoting powder adhesion and thus increasing R a (Figures 11b,11c,13b and 13c).
When the overhang angle increased further (θ > 50°), in addition to the influencing factors discussed above (melt pool sinking and increase of the contact area between the melt pool boundary and powder bed), several other unfavourable factors contributed jointly to the mechanism of powder adhesion, such as border track-induced waviness of the overhang contour, warp deformation, and dross formation. As discussed in Section 4.2.1, the solidified contour of the melt pool of border tracks manifested as the wavy contour of the overhang region (without taking into account the adhered powder particles). It is seen in Figure 10a and c that the waviness at θ = 45°was not very large, indicating that the border track contour was not a significant influencing factor on surface roughness. In contrast, the waviness of the 60°overhang region (Figure 10b and d) was much larger than that of the 45°overhang region, making the border track contour a considerable influencing factor. The wavy contour consisted of a number of undesired protuberances due to overmelting into the loose powder bed. Powder particles adhered to the protuberances forming powder clusters; meanwhile, gaps were manifested between the powder clusters (Figures 10b, d, f and 13d). This was a significant contributor to surface roughness when θ ≥ 60°. Moreover, severe warp deformation and dross also played noticeable roles in contributing to the surface roughness of the overhang region, as shown in Figure 11d and e. Warp deformation and dross were the most crucial factors influencing the surface roughness of the overhang region when the overhang angle increased up to 80°, while the exposure of the inner border contour further enhanced the contour waviness and surface roughness (Figure 11e). The three regimes of powder adhesion related to the overhang angle are summarised in Table 4.
It can be concluded from the preceding discussion that the critical overhang angle is a threshold not only for macro deformations but also for micro shape deviations, which, in the present study, is around 50°. It should be noted that the overhang region with a constant overhang angle of 80°cannot be printed successfully because this large overhang angle results in the collapse of the overhang region during the printing process. But the simulation study on 80°overhang region will benefit the printing of overhang regions with a continuously changing overhang angle (such as horizontal circular channels) which is a challenging task in L-PBF.
The experimental results (Figure 12a) showed that when θ ≤ 55°, the distance from the start point had an insignificant effect on the overhang R a . In other words,  Table 4. Three regimes of powder adhesion related to the overhang angle.

Overhang angle
Regime of powder adhesion 0°≤ θ ≤ 15°(i) A small area of the melt pool boundaries contacts the powder bed resulting in slight powder adhesion on the side walls (ii) The effect of overhang angle is not significant 15°< θ ≤ 50°Powder adhesion is enhanced by: (i) the sinking of the melt pool (ii) the increase of contact area between the melt pool boundary and powder bed θ > 50°Powder adhesion is further promoted and Ra increases sharply due to the combination of: (i) the enhanced sinking of the melt pool (ii) the further increase of contact area between the melt pool boundary and powder bed (iii) large waviness of the overhang contour and the induced powder clusters (iv) severe warp deformation and dross the surface roughness was generally homogeneous on the overhang regions when θ ≤ 55°. On the other hand, however, the experimental results showed a strong trend that R a increased with increasing the distance from the start point of the overhang when θ = 60°. This also indicated that severe warp deformation and dross had a combined effect on R a since those unfavourable factors became more considerable with increasing the length of the overhang region when θ = 60°.

Potential solutions to mitigate surface roughness issue in the overhang region
The volumetric energy density determined the volume and the fluidity of the molten metal. Higher laser energy induced a higher temperature in the melt pool. Firstly, this led to more metal melting, forming a larger melt pool and enlarging the contact area of the melt pool boundaries with the powder bed. Secondly, this resulted in better flowability of the molten metal, causing more molten metal to flow downwards into the gaps between powder particles and promoting the sinking of the melt pool. As discussed in Section 4.3.1, both of these two factors resulted in more powder particles being partially melted or sintered, enhancing powder adhesion and thus increasing R a . Besides, increasing volumetric energy density can also lead to overheating of the powder bed, exacerbating dross formation on the overhang region and further increasing R a .
Therefore, a potential solution to mitigate the high surface roughness issue in L-PBF printed overhang regions is to adjust the process parameters to decrease volumetric energy density (such as decreasing laser power or increasing scan speed) when printing overhang regions. From the experimental results shown in Figure 14a, decreasing laser power was an effective way to decrease R a , which confirmed the previous conjecture. On the other hand, however, R a did not decrease significantly with increasing scan speed (Figure 14b) although this decreased the volumetric energy density. The reason may be that increasing scan speed enhanced the discontinuity of the overhang track, inducing a rougher overhang surface (Le et al. 2020). About the optimisation and feed-forward control of laser power to decrease R a of overhang regions, more studies need to be conducted in the future.

Conclusions
This study investigated the evolution and formation mechanisms of surface morphology in the laser powder bed fused overhang region, taking into account the effects of overhang angle, laser power, and scan speed on surface roughness R a of the overhang region using single-factor experiments and a coupled DEM-CFD simulation model. The main conclusions are summarised as follows.
(1) The melt pool in the overhang region is weakly supported by the powder bed. Molten metal flows downwards into the voids driven by gravity and surface tension-induced capillary effect, inducing sinking of the melt pool. Melt pool sinking is a contributor to powder adhesion as it increases the melt depth compared to the solid region and makes more powder particles being adhered. Melt pool sinking is governed by the overhang angle and volumetric energy density (related to the temperature field of the melt pool). An increasing overhang angle (weakening the support to the melt pool) or an increasing volumetric energy density (improving the fluidity of Figure 14. Effect of (a) laser power (scan speed = 1000 mm/s) and (b) scan speed (lase power = 100 W) on surface roughness R a in overhang regions (θ = 45°, laser power and scan speed referred to overhang border parameters, and the other process parameters are listed in Table 2). the molten metal) can enhance melt pool sinking and vice versa. (2) The evolution of surface morphology is the result of a combination of border track contour, powder adhesion, and shape deviations (warp deformation and dross). The contribution of these influencing factors to surface roughness depends on the overhang angle. Powder adhesion is the main influencing factor to surface roughness, and border contour and shape deviations enhance surface roughness further when θ > 45°. The contours of border tracks manifest as the wavy basal contour of an overhang region. For overhang angle θ ≤ 60°, only outer border tracks contribute to the overhang contour. When θ > 60°, both outer and inner border tracks affect the overhang contour.
(3) The effect of the overhang angle on surface roughness of an overhang region can be divided into three regimes according to the mechanisms of powder adhesion and formation of micro shape deviations. When 0°≤ θ ≤ 15°, the overhang angle has no significant influence on R a . When 15°< θ ≤ 50°, R a increases because of powder adhesion being promoted by (a) increasing contact area between the melt pool boundaries and powder bed, and (b) sinking of the melt pool. When θ > 50°, R a increases sharply under the combined effect of powder adhesion, large wavy contour and the induced powder clusters, and severe warp deformation and dross. When the overhang angle increases up to 80°, warp deformation and dross play the most crucial role in determining R a . (4) R a increases with increasing laser power since the increase of volumetric energy density enlarges the contact area of the melt pool boundaries with the powder bed and promotes melt pool sinking, thus enhancing powder adhesion in the process.
In laser materials processing, the melt pool shape and isotherm contours are strongly dependent upon the shape of the equivalent volumetric heat source. For a Gaussian laser beam, taking into account the attenuation in the depth direction (i.e. into the powder bed and substrate) dictated by the Beer-Lambert law, the distribution of laser power density inside the material can be expressed as (Feng et al. 2017): Q = I 0 a · exp (−2r 2 /r 2 0 ) · exp (−az) ( A.1) where Q, I 0 , and α are the laser power density inside the material, peak laser intensity, and attenuation coefficient, respectively. We seek to define the boundaries of the 3D volume that can be approximated as an equivalent heat source in the model. To this end, the volumetric heat source is assumed to reach its boundaries when the laser power density attenuates to a certain value. This threshold boundary value of laser power density (Q B ) is equal to the value of the surface power density at a radial distance equal to the spot radius of the laser, and can be determined by: As mentioned before, the volumetric heat source is assumed to have a power density of Q B at its bounding surface. If r B and z B define the locus of points that define the volumetric heat source in the cylindrical coordinate system, we have: Combining Eqs. A.2 and A.3: I 0 a · exp (−2) = I 0 a · exp (−2r 2 B /r 2 0 ) · exp (−az B ) (A.4) Simplifying Eq. A.4, we obtain: 2r 2 B /r 2 0 + az B = 2 ( A.5) Further, we note that the optical penetration depth is: Combining Eqs. A.5 and A.6, we obtain: Finally, combining Eqs. A.5 and A.7, we obtain: Eq. A.8 describes an equation determining the bounding surface of the 3D volumetric heat source. In other words, the volumetric heat source is comprised of all points (r, z) such that r ≤ r B and z ≤ z PD . This 3D shape was used as the volumetric heat source (Eq. 5) in the model, and the incident laser energy was assumed to be generated uniformly within this 3D volume.