Laser spot size and scaling laws for laser beam additive manufacturing

Abstract

Laser powder bed fusion (L-PBF) additive manufacturing (AM) requires the careful selection of laser process parameters for each feedstock material and machine, which is a laborious process. Scaling laws based on the laser power, speed, and spot size; melt pool geometry; and thermophysical properties can potentially reduce this effort by transferring knowledge from one material and/or laser system to another. Laser spot size is one critical parameter that is less well studied for scaling laws compared to laser power and scan speed. Consequently, single track laser scans were generated with a spot size (D4σ) range of 50 μm to 322 μm and melt pool aspect ratio (depth over spot radius) range from 0.1 to 7.0. These were characterized by in-situ thermography, cross-sectioning, and optical microscopy. Scaling laws from literature were applied and evaluated based on melt pool depth predictions. Scaling laws that contain a minimum of three dimensionless parameters and account for changing absorption between conduction and keyhole mode provide the most accurate melt pool depth predictions (< 35 % difference from experiments), which is comparable to thermal simulation results from literature for a select number of cases.

Graphical Abstract:

1. Introduction

Laser powder bed fusion (L-PBF) additive manufacturing (AM) requires the careful selection of laser process parameters for each feedstock material and original equipment manufacturer (OEM) to produce high quality parts. The selection of laser process parameters for structural alloys is often an intensive effort, multi-objective optimization process for maximizing density, build rate, and mechanical properties while reducing residual stress and anisotropy. Transferring knowledge from an optimal set of laser process parameters to a new material or machine at a minimum requires a knowledge of the effect of the thermophysical properties and laser process parameters on the melt-pool morphology. Melt pool morphology, particularly width and depth measured from ex-situ cross-sections, are widely used in the development and characterization of AM process maps by relating to process parameters such as scan speed and laser power. For example, Yadroitsev and Smurov [1] demonstrated that processing parameters have stability zones that form continuous, defect free tracks, and Beuth et al. [2] demonstrated process mapping for multiple AM systems and alloys. Melt pool dimensions are also used directly to develop simple geometrical models that have been shown to be valuable to avoid lack-of-fusion porosity [3] as well as optimizing process parameters for faster build rates [4].

There are various bases for comparing different laser process parameters. Volumetric energy density is commonly used to explain the effects of varying process parameters such as power, speed, hatch spacing, and powder layer thickness on the final part performance. Bourell et al. [5] showed this most clearly for correlating density and tensile strength to process parameters of PBF polyamide. This works well in some cases; however, it fails to capture melt-pool physics and track morphology [6] such that it lacks enough rigor to transfer process parameters from one material and/or laser system to another. Similarly, Mukherjee et al. [7] have shown the ratio of laser power to scan speed normalized by a reference ratio, termed ‘non-dimensional heat input’, can be used to explain changes in melt pool morphology, cooling rate, density, thermal strain, and hardness. They also show how other dimensionless numbers that incorporate thermophysical properties (i.e., Peclet, Marangoni, and Fourier) could be used to optimize specific outcomes of the additive process [7]. This was successful for changing laser power with a fixed scan speed and laser spot size [7]. However, there is no comprehensive scaling relationship from one material and laser system to another. Two important things missing from the volumetric energy density and non-dimensional heat input are the laser spot size and thermophysical material properties. Another approach, referred to as scaling laws, focuses solely on laser process parameters and thermophysical material properties using dimensionless analyses with a simple thermal model description of the melt pool. These scaling laws for laser welding and L-PBF specifically incorporate laser power, speed, and spot size along with material properties and have been shown to unify melt pool depth and aspect ratio onto single curves for various process parameters and materials (e.g., [8]). They do not account for scanning strategy process variables such as hatch spacing or other process variables such as powder layer thickness. However, because they directly incorporate the melt-pool morphology and some of the physics governing the process, they are arguably a good starting point for comparing and transferring laser process parameters to new materials and/or laser systems.

There are several works demonstrating scaling laws. Hann et al. [8] compared laser welds on stainless steel, V, Ti, and Ta. They introduced a relationship between the ratio of the melt pool depth to the laser spot size and the ratio of deposited energy normalized by the enthalpy of melting. The later ratio is referred to as normalized enthalpy. The relationship between the ratio of melt pool depth to spot size and normalized enthalpy provided a single curve to describe the data from the four materials up to the transition from conduction to keyholing mode after which data from different materials separated. An additional term, which was a function of the Fourier number, was needed to collapse this relationship to a single curve from conduction to keyhole mode, particularly for Ta. King et al. [9] applied normalized enthalpy to L-PBF of stainless steels showing that it provides a meaningful way to compare tracks produced under different laser process conditions, and it is useful for identifying the transition from conduction to keyhole mode. Rubenchik et al. [10] built on these efforts to produce universal functions for melt pool depth, width, and length from simulations and experiments, and Ye et al. [11] extended the scaling laws to include changes in absorption from conduction to keyhole mode. Lastly, Fabbro [12] has shown that dimensionless analysis used to describe laser welding can be applied to L-PBF, and that normalized enthalpy is one of several possible dimensionless parameters useful for scaling laws. In all this work, laser power and scan speed have been the primary variables with limited comparison to the effect of laser spot size variation.

It is largely understood that laser power and speed have the greatest effect on the melt pool morphology compared to other laser process parameters. For example, King et al. [13] and Ma et al. [14] have shown this with thermal simulations through design of experiments and statistical measures. As such, the effect of the laser spot size is less rigorously studied. However, laser spot size is critical when comparing results from AM machines with different laser systems or on machines where laser spot size can be intentionally varied. Furthermore, Francis [15] has shown that the process space in L-PBF that avoids keyholing and bead up melt pools can be expanded by incorporating laser spot size in process mapping. This work focuses on spot size effects on the melt pool morphology and the use of scaling laws to describe the effect on melt pool depth.

Single track laser scans covering conduction to keyholing mode were employed to investigate spot size effects. These were performed on bare plates of nickel superalloy 625 (IN625) to eliminate the effect of the powder. Previous experiments showed the existence of a single layer of powder has minimal effect on melt pool geometry [16]. Experiments were performed on two different machines with fiber lasers that use different methods to change the laser spot size. A limited number of laser power and speed combinations were tested. The primary laser power and speed combination was 195 W and 800 mm s⁻¹ for which the D4σ¹ spot size (spot diameter) was varied from 50 μm to 322 μm with more than 18 different sizes. Four scaling relationships were evaluated using the single-track experiments. In particular, the melt pool depth prediction from different scaling laws was compared against experiments and critically discussed.

2. Materials and Methods

2.1. AM Machines and Spot Size Determination

Two different laser powder bed fusion machines were used in this study: a commercial EOS M270² and a custom research platform named the additive manufacturing metrology testbed (AMMT), which is described by Lane et al. [17]. Both systems use an Yb-fiber laser; however, the laser spot diameter is set through different mechanisms on the AMMT and EOS 270. The AMMT uses a dynamic linear translating z-lens (LTZ) to maintain uniform laser focus across the build platform or flat-field scanning. The LTZ is mounted and aligned on a second linear stage, which adjusts the static laser focus position and spot size [17]. The EOS 270 uses an f-theta lens to perform flat-field scanning and an adjustable defocusing lens after the laser collimator to adjust the spot size. The AMMT laser spot size was measured by first attenuating the power by pulsing at 1/1000 duty cycle and placing neutral density filters in the collimated section of the optical path. The power distribution is then measured at the build plane by exposing the laser on a charge-coupled device (CCD) array [18]. The measured power distribution was later verified using a commercial off-the-shelf laser beam sampler. The EOS 270 spot size was estimated based on the manufacturer’s literature, technical correspondence with the manufacturer, and machine settings. The estimates were shown to be reasonable based on a comparison of single-track laser scan cross-sections between the EOS 270 and AMMT. The spot size estimate for the EOS 270 was determined as follows. The spot diameter for contour scanning mode was assumed to be 80 μm. When the defocus unit is engaged, it changes the spot diameter from 100 μm to 500 μm in a linear fashion for defocus unit settings from 0 to 9. The largest defocus unit setting used in this study was 5, which produces an approximate spot diameter of 322 μm. In addition to changing the spot size via the defocus unit on the EOS 270, the substrate was moved upward from the recoating plane to a maximum of 4 mm, which increases the spot size. In order to determine the spot diameter, D4σ, for a given Z-offset from the recoating plane, the beam caustic must be known. The beam caustic, shown in Figure 1a, was estimated according to Eqn. (1), where λ is the wavelength (1070 nm), D₀ is the beam waist diameter, f_R is the Rayleigh length, and f_d is the Z-offset from the beam waist:

$$D4\sigma (f_d)=D_0\sqrt{1+{(\frac{f_d}{f_R})}^2},$$ (1)

$$f_R=\frac{\pi {D_0}^2}{4\lambda }.$$ (2)

Figure 1.

(a) Estimated beam caustic for the EOS 270, f_d is the distance from the beam waist, D₀ (b) estimated spot diameter, D4σ, versus the Z-offset from the recoating plane for four different defocus settings: contour (focused) mode and defocused at settings 0, 2, and 5. A positive Z-offset moves the plate above the recoating plane toward the f-theta lens. Roman numerals (i) through (v) correspond to scenarios of defocus setting and Z-offset from the recoating plane, Z, on both figures: (i) Contour, Z = 0, (ii) Defocus = 0, Z = 0, (ii) Defocus = 0, Z = 2 mm, (iv) Defocus = 2, Z = 0, (v) Defocus = 5, Z = 0.

The beam waist, D₀ = 78 μm, was assumed to be slightly smaller than the focused beam spot diameter at the recoating plane, D4σ = 80 μm, because the beam waist is positioned 0 to 1 mm below the recoating plane. The effect of the defocus unit setting and Z-offset from the recoating plane on the spot diameter were combined by first calculating the equivalent Z-offset for a given defocus unit setting spot diameter by solving for f_d in Eqn. (1). Next, the physical Z-offset of the substrate from the recoating plane was added to the equivalent Z-offset from the defocus unit setting. The spot diameter was then calculated based on the combined Z-offset using Eqn. (1). The spot diameter versus Z-offset from the recoating plane for a few different starting beam sizes (different defocus unit settings) is shown in Figure 1b.

2.2. Single Tracks and Melt Pool Measurements

Single scan laser tracks were made on nickel superalloy 625 (IN625) substrates. The substrate size, track length, and track spacing are shown in Figure 2. Laser tracks were cross-sectioned in the approximate center of the track and metallographically prepared. The solidified melt-pool boundary was revealed by immersion etching with Aqua Regia. Each track was measured twice by either preparing both halves of the plate or regrinding, polishing, and etching after making the first measurement. Optical images were taken on a Zeiss LSM 800 with a 50× magnification, 0.186 μm pixel size, and bright field mode. The melt pool depth was measured from the substrate surface to the bottom of the melt pool. This neglects any humping that may occur on the top surface of the melt pool. The melt pool width was measured from edge to edge where the melt pool meets the unmelted substrate surface. The melt pool length was determined with in-situ thermography using an infrared indium antimonide (InSb) camera. A complete description of the camera setup and measurement is described in [19, 20]. Briefly, a custom door with a viewport was made for the EOS 270 to allow for the infrared camera to view a small region of the build plane at an incline of 46.3° from the surface normal. The camera was equipped with a 50 mm short-wave infrared lens and optical filter with a bandwidth of 1350 nm to 1600 nm. The field of view is approximately 12.96 mm × 6.82 mm (360 pixels × 128 pixels). The camera is calibrated using a variable temperature blackbody source. The calibration procedure maps digital signals from the camera to the setpoint temperature of the blackbody source using the Sakuma-Hattori equation. The melt pool length is measured from the radiance temperature (e.g., apparent temperature) line profile along the center of the melt pool in the laser scanning direction assuming that the inflection point (or minimum of the second derivative) in the measured radiance temperature profile behind the laser corresponds to solidification at an assumed 1290 °C for IN625. The length is measured from this point behind the laser to the intersection of the same radiance temperature value on the front side of the laser. The cooling rate is determined over the range of 1290 °C to 1000 °C. Each frame is analyzed while the laser scan is in the field of view to produce and average and standard deviation for one track. It is noted that the melt pool length measurement does not depend on the assumption of solidification at 1290 °C while the cooling rate depends on this assumption and therefore is an approximation. Most substrates were in the as-received condition, Figure 2(b). A couple plates were ground with 320 grit SiC paper, Figure 2(c), to assess the effect of the difference in surface finish from prior work [20]. Laser tracks for the two different surface finishes were similar based on four tracks (5 measurements for ground and 8 measurements for as-received) produced at the same settings for each surface finish. Laser tracks on the EOS 270 at 195 W, 800 mm s⁻¹, and D4σ = 100 μm had a (mean ± one standard deviation) melt pool depth, width, and length of (94.3 ± 4.5) μm, (140.6 ± 10.5) μm, (798.7 ± 34.3) μm for a ground substrate and (89.4 ± 5.6) μm, (131.5 ± 7.48) μm, (788.7 ± 75.6) μm for as-received substrate, respectively. The two surfaces are treated as identical for the purposes of this study.

Figure 2.

(a) Schematic of bare plate and single track laser scans, (b) example optical image of laser tracks on an as-received plate, (c) example optical image of laser tracks on a ground plate with 320 grit SiC paper. The thickness of tracks is due to difference in laser settings and not the surface type.

A summary of the laser parameters used in this study is provided in Table 1. Again, the main emphasis is on the effect of the laser spot size, and so most laser scans were made at a laser power and scan speed of 195 W and 800 mm s⁻¹, respectively. Ten single track laser scan experiments produced on the EOS 270 from Lane et al. (2020) were included in the present data set. These tracks were produced and measured under similar conditions to the present work. In total, the data set contains 80 single track laser scans.

Table 1.

Summary of single track laser scan parameters.

Machines	Plate Surfaces	Laser Powers (W)	Laser Scan Speeds (mm s−1)	Spot Diameter Range (μm)
AMMT, EOS 270	As-received, 320 grit	122, 150, 179.2, 195	200, 400, 800, 1200	50 to 322

3. Theory

Scaling laws for laser welding and laser based additive manufacturing unify the effect of laser process conditions on the resultant weldment or melt pool across different materials and machines. There are several proposed scaling relationships that are based on simple conductive models describing the melt pool. The following is a brief description of the different scaling relationships used to analyze experiments in this work. King et al.[9] demonstrated the onset of keyholing could be predicted for different materials using a normalized enthalpy term, ΔH/h_s, which was first introduced by Hann et al. [8]. The normalized enthalpy is a function of the absorption, A, laser power, P, enthalpy of melting, h_s, thermal diffusivity, D, laser scan speed, u, and laser spot size, σ, as shown in Eq. (1):

$$\frac{\mathrm{\Delta }H}{h_S}=\frac{AP}{h_S\sqrt{\pi Du{\sigma }^3}}.$$ (3)

King et al. [9] defined the enthalpy of melting based on the melting temperature, T_m: h_s = ρcT_m, where ρ is the material density and c is the specific heat capacity. Here we adopt the more general calculation, which includes the initial substrate temperature, T₀, as shown in Eq. (4):

$$h_s=\rho c(T_m-T_0).$$ (4)

Recall that the spot diameter = 4σ or the spot radius = 2σ. King et al. [9] showed that for a plot of the melt pool depth to laser spot size ratio, d/σ, versus the normalized enthalpy there is a transition at ΔH/h_s ≅ 30 between conduction and keyholing. Rubenchik et al. [10] built on this work by introducing a thermal diffusion depth³, δ:

$$\delta ={\left(\frac{D\cdot \sqrt{2}\sigma }{u}\right)}^{\frac{1}{2}}.$$ (5)

Rubenchik et al. [10] demonstrated that for a plot of the melt pool depth to thermal diffusion depth ratio, d/δ, versus normalized enthalpy, ΔH/h_s, brought experimental data for steel, Ti-6Al-4V, and IN625 onto a single, linear curve with a positive slope ≅ 0.25. They also proposed a relationship of melt pool depth, width, and length as a function of normalized enthalpy and the Peclet number [10]. This relationship was determined as a set of universal functions from a regression analysis of simulations, which is included in Appendix A.

Ye et al. [11] combined the work of King et al. [9] and Rubenchik et al. [10] with the additional sophistication of a non-constant laser energy absorption coefficient. Ye et al. [11] introduced the normalized thermal diffusion length, $L_{th}^{*}$, which can be shown to be thermal diffusion depth from Rubenchik et al. [10] normalized by the spot radius. This also results in an expression of the Peclet number as shown in Eqn. (6).

$$L_{th}^{*}=\frac{\sqrt{\frac{D\cdot 2\sigma }{u}}}{2\sigma }=\sqrt{\frac{D}{2\sigma u}}=\frac{1}{\sqrt{Pe}}$$ (6)

Ye et al. [11] used a formula for normalized enthalpy, Eqn. (7), which is different than the formula in Eqn. (3). This is the result of a derivation that differs from Hann et al (2011).

$$\beta =\frac{AP}{\pi h_s\sqrt{Du{(2\sigma )}^3}}.$$ (7)

Ye et al. [11] proposed that there is a linear relationship between the depth to spot radius ratio and the normalized thermal diffusion length multiplied by normalized enthalpy, where K = 0.6 is an empirical constant for all materials and process conditions:

$$\frac{d}{2\sigma }=K\beta L_{th}^{*}=K\left(\frac{AP}{\pi h_s\sqrt{Du{(2\sigma )}^3}}\right)\sqrt{\frac{D}{2\sigma u}}.$$ (8)

Furthermore, Ye et al. [11] accounted for a non-constant absorption, A_e, using an empirical fit to calorimetry-based laser energy absorption measurements for different laser parameters, Eqn. (9)⁴, where A_m is the baseline absorption during conduction mode, and the absorption increases until it saturates at 0.70:

$$A_e=\left\{\begin{array}{cc}{A}_m& \frac{d}{2\sigma }\le 1\\ 0.70(1-e^{-0.66{\beta }_{A_m}{L}_{th}^{*}})& \frac{d}{2\sigma }>1\end{array}.\right.$$ (9)

In this case, Eqn. (8) is calculated using ${\beta }_{A_e}$ and the subscripts on β indicate whether a constant A = A_m or non-constant A = A_e is used to calculate β according to Eqn. (7). The scaling law from Ye et al. [11] was shown to provide a linear response for several alloys over a variety of laser process conditions.

Lastly, Fabbro [12] takes a different approach, which is based in laser welding literature. Fabbro [12] shows that there is a linear relationship according to Eqns. (10–12) where R = d/4σ, κ is the thermal conductivity, T_v is the metal boiling or vaporization temperature, and m = 2.4 and n = 3 are empirical constants for g(Pe) = mPe + n, 2 ≤ Pe ≤ 10:

$$\frac{AP}{4\sigma R}=a+b(u4\sigma ),$$ (10)

$$a=n\kappa (T_v-T_0),$$ (11)

$$b=\frac{m\kappa (T_v-T_0)}{2D}.$$ (12)

Fabbro [12] showed that this scaling law provides a linear relationship that agrees with the predicted slope and intercept for different sized keyholes in steels. In addition, Fabbro [12] showed that Eqn. (10) and the use of dimensionless parameters for scaling laws such as the depth over spot size, normalized enthalpy, and the melt pool depth over thermal diffusion depth can all be derived through a dimensionless analysis using the Vaschy-Buckingham π-theorem [21].

While laser spot size is integral to the formulation of these scaling laws, there is limited data verifying their sensitivity to this parameter. Additionally, the evaluation of these scaling laws has been based on collapsing data from multiple materials and different laser parameters onto a single curve. In the present study, these scaling laws will also be evaluated based on how well they predict melt pool depth, which will provide additional insight into their usefulness and limitations for making quick decisions on processing parameters. The predicted depth for Rubenchik et al. [10], Ye et al. [11], and Fabbro [12] are listed in Eqns. (13–15), respectively. The constant, 0.25, in Eqn. (13) comes from Fabbro’s analysis of Rubenchik’s data. It should be noted that all three predictions rely on empirical constants in some form or another. A table of the thermo-physical properties used in calculations is give in Table 2.

Table 2.

Thermo-physical properties and conditions used for IN625. Diffusivity was calculated: D = κ/ρc_p. The absorption coefficient is specific to each analysis and is given for each result.

T0 (K)	300.15	ρ (kg m−3)	8440
Tm (K)	1563.15	κ (W m−1K−1)	28.78
Tv (K)	3500	cp (J kg−1 K−1)	719.92

$$d=0.25\cdot \delta \frac{\mathrm{\Delta }H}{h_s}$$ (13)

$$d=2\sigma K{\beta }_A{L}_{th}^{*}$$ (14)

$$d=AP/\kappa (T_v-T_0)(mPe+n)$$ (15)

4. Results

4.1. Spot size effects on melt pool dimensions

The single-track laser scan data including representative micrographs is presented first followed by the evaluation of different scaling laws to describe the data. Representative cross-sectional micrographs for changing spot diameter using the defocus unit on the EOS 270 are shown in Figure 3. For reference, the contour scan is performed with a spot diameter of 80 μm, and the fill or hatching scan is performed with a spot diameter of 100 μm for the EOS IN625 parameter set. In these conditions at the default laser power and scan speed, the process is in keyhole mode, which we define when the measured melt pool cross-section depth to spot radius is greater than one. A more common definition in literature is when the melt pool depth to half width is greater than one [22]. These two definitions are nearly identical, and the ratios only significantly deviate for values greater than one (see Appendix B). The results from the AMMT and changing the Z-offset on the EOS 270 are shown in Appendices C and D, respectively, and show similar melt pool sizes to those shown in Figure 3. In addition to changes in the spot diameter, changes to laser power and scan speed have a significant effect on the melt-pool cross-section. A few conditions of lower laser power and different scan speeds were conducted. Representative micrographs showing a select set of the parameters, which includes the most extreme keyholes in the data set, are shown in Figure 4.

Figure 3.

Representative cross-sectional micrographs of single track laser scans on a bare substrate for increasing spot diameter by changing the defocus setting from (a) to (h). The laser power and scan speed were constant at 195 W and 800 mm s⁻¹, respectively. The sample was etched with Aqua Regia.

Figure 4.

Representative micrographs for a fixed laser spot diameter of 100 μm with changing laser power and scan speed. The sample was etched with Aqua Regia.

The measurements of the single-track experiments are shown in Figure 5. There are eight classes of data from seven different laser power and speed combinations on the EOS 270 machine and one laser power and speed combination on the AMMT. There is a similar response for the EOS and AMMT machines at the same laser power and scan speed for melt pool depth and width, which supports the accuracy of the estimated spot diameter based on the manufacturer’s specifications. There is clearly a dependence on the melt pool geometry and surface cooling rate based on laser power, scan speed, and spot diameter. Due to the nature of keyholing or laser drilling, the melt pool depth shows the greatest dependence on the laser parameters compared to width and length. The expected behavior of increasing melt pool depth for higher power to speed ratios is observed. Similarly, the expected behavior for increasing melt pool depth for decreasing spot diameter is observed. For the laser power and scan speed of 195 W and 800 mm s⁻¹, there is a transition point in the depth, width, length, and cooling rate between a spot diameter of 100 μm and 150 μm. This corresponds to the transition between conduction and keyholing based on the depth to laser spot radius ratio. The dependence of the melt pool geometry on process parameters and thermo-physical properties should be captured in the scaling laws, which is presented next.

Figure 5.

(a) Melt pool depth versus spot diameter, (b) melt pool width versus spot diameter, (c) melt pool length from in-situ thermography versus spot diameter, and (d) cooling rate from in-situ thermography versus spot diameter. Data in (c) and (d) are the mean ± one standard deviation over the duration of each track while it is in the camera field of view. Thermography measurements were only made on the EOS machine.

4.2. Scaling laws

First, the data was analyzed using the normalized enthalpy plot by King et al. [9], Figure 6. This is not necessarily a scaling law, but it is inspired by the scaling law by Hann et al. [8], and it the starting point for the scaling laws by Rubenchik et al. [10] and Ye et al. [11]. The analysis by King et al. [9] is most useful for determining the threshold for keyholing for different laser powers, speeds, and beam sizes. The threshold in 316L stainless steel occurred at a normalized enthalpy of approximately 30 [9]. The same threshold is observed in this data on IN625. Data beyond this threshold is scattered, which indicates that using these two parameters does not collapse the data into a single curve. At least one additional dimensionless number is needed according to Fabbro [12], who has suggested that at least three dimensionless parameters are needed for a scaling law. The scaling laws by Rubenchik et al. [10], Ye et al. [11], and Fabbro [12] all use three dimensionless parameters.

Figure 6.

Melt pool depth over spot size versus normalized enthalpy following King et al. (2014). A constant absorption of 0.4 was used.

The results from Rubenchik et al. [10], Ye et al. [11], and Fabbro [12] scaling laws are shown in Figure 7. The scaling law from Ye et al. [11] was applied twice: first using the absorption coefficient and function from their study (referred to as Ye et al. [11]), and second using a modified absorption coefficient and function that reduces the difference between the melt pool depth prediction and experiment (referred to as Modified Ye et al. [11]). The modification also provides better agreement with absorption coefficients measured by Deisenroth et al. [23]. Each scaling law plot in Figure 7 has a linear regression with prediction intervals at 95 % confidence. A summary of the linear regressions analysis is given in Table 3 with the expected slopes and intercepts based on regression analyses or theory. The results presented in Figure 7 and Table 3 are discussed next along with an explanation for the absorption coefficients for each scaling law.

Figure 7.

Scaling law plots for (a) Rubenchik et al. (2018) [10], (b) Fabbro (2019) [12], (c) Ye et al. (2019) [11], and a modified Ye et al. (2019) [11]. Important key parameters are listed on the plots such as absorption, A, and vaporization temperature, T_v. The solid line is a linear fit and the dotted lines are prediction intervals at 95% confidence

Table 3.

Linear regression coefficient results comparing the abscissa and ordinate variables associated with each scaling law and comparison to expected coefficient.

	Slope (upper and lower value at 95% confidence)	Intercept (upper and lower value at 95% confidence)	R2	expected slope	expected intercept
Rubenchik et al. (2018) [10]	0.256 (0.247, 0.266)	−0.270 (−0.493, −0.047)	0.948	0.25	n/a
Fabbro (2019) [12]	2.94 (2.81, 3.07) × 1010	−3.48 (−5.10, −1.86) × 105	0.930	2.33 × 1010	2.76 × 105
Ye et al. (2019) [11]	0.631 (0.599 0.663)	0.491 (0.407, 0.576)	0.912	0.6	0
Modified Ye et al. (2019) [11]	0.518 (0.504, 0.532)	0.151 (0.097, 0.205)	0.971	0.6	0

Plotting the abscissa and ordinate variables defined by the scaling law in Rubenchik et al. [10] in Figure 7(a) shows a linear trend, especially compared to Figure 6. The slope (and upper, lower value at 95% confidence) from the linear fit, 0.256 (0.247, 0.266), matches the expected value of 0.25 (no confidence intervals given) [12]. The scaling law variables by Fabbro [12] also show a linear relationship in Figure 7(b). Here, it is noted that the most extreme keyholing data points are closest to the origin; whereas, they are furthest from the origin in the other scaling law plots. Fabbro’s model is based in laser welding, which primarily operates in keyhole mode; therefore, a higher absorption coefficient of 0.8 was used. Another estimated variable is the vaporization temperature, T_v = 3500 K, which is the value Fabbro uses for steel [12]. Both the absorption coefficient and vaporization temperature can be adjusted within reason to bring the empirical slope and expected slope in agreement (lower absorption coefficient and/or higher vaporization temperature). Thus, the slight disagreement in Table 3 for Fabbro [12] should not be emphasized. However, the intercept remains negative. This could be caused by a limited number of data points with deep keyholes; whereas, Fabbro’s data had a higher percentage of data points with deep keyholes. One final source of uncertainty comes from the Peclet function, which has a range of applicability (see Eqns. (10–12)). The Peclet number range for this data is 2.1 to 27.2, which is outside the range of 2 to 10 for Eqns. (10–12) [12]. The highest Peclet numbers occur for the largest spot diameter in this data set, and this is again a possible reason why the intercept does not agree with the predicted value.

The scaling law by Ye et al. [11] in Figure 7c also provides a linear trend. Qualitatively, the data appears to be more compact; however, the fit also has the lowest R² value (0.915 compared to 0.930 and 0.948) along with a shift in the curve at the keyhole transition. The absorption function used by Ye et al. [11] has three coefficients that determine the absorption in conduction mode or baseline (A_m = 0.28), the rate at which the absorption increases in the transition from conduction to keyholing (−0.66), and the saturation or maximum absorption with further keyholing (0.70). The modified Ye et al. [11] case in Figure7c has a higher baseline (A_m = 0.5) and maximum absorption (0.8) along with doubling the exponential coefficient (−1.32) so that the absorption transitions from the baseline to the maximum more quickly. It will be shown in the next section that the doubling of the exponential coefficient along with the increases in absorption coefficients improves the melt pool depth prediction. In both Ye et al. [11] results, the slope is near the expected value of 0.6. The modified Ye et al. [11] result provides an intercept much closer to zero. Like the other models, adjusting the absorption coefficient/function within reason can provide a match with the expected slope.

4.3. Melt pool depth predictions

While the literature and this study so far have shown that the scaling law variables have a highly linear relationship for a range of processing parameters and materials, evaluation of the scaling laws based on melt pool depth predictions is lacking. This section presents the capability of these scaling laws for depth predictions. The melt pool depth, length, and width predictions from the universal equations from Rubenchik et al. [10] are evaluated separately in Appendix A. The melt pool depth prediction for Rubenchik et al. [10] presented in this section is based on Eqn. (13). The predicted depth from each scaling law versus the measured depth is shown in Figure 8. The percent difference between the predicted and measured depth versus the measured depth to spot radius ratio is shown in Figure 9. The reader is referred to Eqns. (11–13) and the corresponding Figure 7 for the calculation of depth predictions.

Figure 8.

Melt pool depth prediction versus measured depth using scaling laws from (a) Rubenchik et al. (2018) [10], (b) Fabbro (2019) [12], (c) Ye et al. (2019) [11], and (d) a modified Ye et al. (2019) [11]. See corresponding Figure 7.

Figure 9.

Percent difference between the predicted and measured melt pool depth versus the depth to spot radius ratio using scaling laws from (a) Rubenchik et al. (2018) [10], (b) Fabbro (2019) [12], (c) Ye et al. (2019) [11], and (d) a modified Ye et al. (2019) [11]. See corresponding Figures 5 and 6. A positive value indicates there is an over prediction, and a negative value indicates there is an underprediction.

The predicted depth plots reveal additional information about the accuracy of the scaling laws. Despite assuming a constant absorption, the Rubenchik et al. [10] scaling law provides more accurate predictions compared to Fabbro [12] and Ye et al. [11], Figure 8. The maximum percent difference is higher than 50 % for data below an aspect ratio (depth to spot radius) of 1, and there seems to be a trend in the percent difference plot suggesting there is something missing from the prediction model, Figure 9a. It is also interesting that the Fabbro [12] model is most accurate for low aspect ratio data below keyholing when it was mostly developed for keyhole mode laser welding. There is a clear trend in the percent difference plot for Fabbro [12] with over predicted depths at the smallest aspect ratios (50 %) and under predicted depths at the highest aspect ratios (−90 %), Figure 9a. This is not entirely surprising given that the scaling law slope and intercept did not agree with the expected values, Table 3. Changing the vaporization temperature and absorption coefficient shifts the data up and down; however, it doesn’t influence the trend. The result for Ye et al. [11] at first glance looks poor for the majority of the data, Figure 9c, and there is a large percent difference between the predicted and measured. However, the percent difference is consistent. This leads to the modified Ye et al. [11] result.

The modified Ye et al. [11] scaling law included three adjustments to the absorption as described in the previous section. Further description is given here. The difference is most succinctly summarized in the equation insets on Figure 7c and d. The baseline, A_m, was increased, 0.28 to 0.50, until the percent difference was clustered around zero percent difference for the low aspect ratio data. Next the maximum absorption was increased, 0.7 to 0.8, so that the data at high aspect ratios clustered around zero percent difference. Last the exponential coefficient was adjusted, −0.66 to −1.32, to flatten out the percent difference data in the transition. A direct comparison of the absorption function with coefficients from Ye et al. [11] and the modified function as described above is also made in Figure 10. The absorption is plotted versus the normalized enthalpy multiplied by the normalized thermal diffusion depth,${\beta }_{A_m}{L}_{th}^{*}$. While all the adjustments were made to provide better agreement between the predictions and measurement, they can also be justified. The baseline and maximum absorption coefficients are closer to those measured by Deisenroth et al. [23], which start at a range 0.34 to 0.40, increase quickly to 0.60, and steadily increase further to a maximum value of 0.90 for increasing laser power from 50 W to 300 W with a fixed laser spot diameter, 65 μm, and scan speed, 500 mm s⁻¹. These absorption measurements were made via an energy balance on the same AMMT system on IN625 bare plates. The increase in the exponential coefficient is justified by the observation that the absorption data in Ye et al. [11] shows a sharper increase than the regression used by the authors. For the modified Ye et al. [11] scaling law results, the predictions are in best agreement with the experiments. Even so, the percent difference is as high as ±35 %.

Figure 10.

Calculated absorption values as a function of normalized enthalpy, ${\beta }_{A_m}$ multiplied by the thermal diffusion length, $L_{th}^{*}$. Each data point represents a single track experiment for which the absorption was calculated based on the laser process parameters, thermophysical properties, and melt pool depth. See Eqn. (7) and Figure 7.

6. Discussion

6.1. Missing physics in the scaling laws

The single most influential parameter in the scaling laws for melt-pool depth predictions is the absorption coefficient. It is clear from literature that absorption increases from conduction to keyholing. Rubenchik et al. [10] provides a scaling law that appears to be less sensitive to the fact that the absorption is not constant. Even so, it seems prudent to account for the non-constant absorption in scaling laws. Ye et al. [11] have shown that a non-constant absorption could be accounted for across materials and laser parameters using Eqn. 9. They also note that there are likely still some missing physics, as there is some dependence on scan speed that is unaccounted for. Fabbro [24] has postulated about different functional forms for a non-constant absorption based on the melt-pool aspect ratio. Fabbro [24] also proposes that a non-constant absorption is described by the melt-pool inclination angle at the front of the advancing melt-pool. This seems like a more direct way to account for a non-constant absorption since it is the reflection of laser traces within the melt-pool depression that is increasing absorption. We hypothesize that differences in the inclination angle during keyhole mode for similar ${\beta }_{A_m}{L}_{th}^{*}$ terms could explain the scatter in data for the Ye et al. [11] scaling laws at larger aspect ratio melt-pools. However, measuring the inclination angle requires in-situ synchrotron experiments, and it is not clear how to predict it in a scaling relationship that accounts for material properties and laser parameters. It is also apparent that temperature dependent thermal-physical properties of alloys up to vaporization temperature are difficult to come by [11]. It is a simplification to use the melting temperature. Fabbro [12] uses the vaporization temperature; however, there is a lack of data for common alloys compared to the melting temperature. Keyholing is driven by a recoil pressure; therefore, there should be some dependence on the environment pressure (e.g., inert atmosphere versus vacuum). This was explored and demonstrated by Calta et al. [25] where they showed that a difference in the vapor depression aspect ratio was observed with synchrotron experiments. It was reasoned that the different environment pressures change the boiling temperature and melt pool surface temperature, which change the melt pool surface tension. Thus in-situ experiments that measure absorption and the melt pool shape seem primed to better inform scaling laws and physics-based models.

6.2. Comparison with thermal simulations

The best-case scaling law required updating the absorption function, adjusted Ye et al. [11], which resulted in ± 35 % difference. To put this in perspective, a comparison to computational thermal simulation is provided. The melt pool depth measurements from the recent AM Benchmark 2018 challenge [20] to predict melt pool morphology for single track laser scans on IN625 produced on the EOS 270 is given Table 4. The blind predictions stemming from computational simulations of varying complexity, and the corresponding scaling law predictions are given in Table 5. The scaling law predictions have similar or better, Rubenchik et al. [10] and modified Ye et al. [11], accuracy than the thermal simulations. As noted by the depth over spot radius ratio, all three cases are in transition or keyhole mode. The thermal simulations lack some physics to describe keyholing mode, as was the case for Rubenchik et al. [10] universal functions fit to thermal simulations (Appendix A). The scaling laws rely on empirical constants that come from a range of data from conduction to keyholing but were developed with the keyholing in mind. Hence, they perform as well as thermal simulations under keyhole conditions. The scaling laws do not perform well in conduction mode as shown in Figure 9a. We hypothesize that the thermal simulations would provide more accurate predictions for laser parameters that result in conduction mode processing. Another shortcoming of the scaling laws is that they do not provide a prediction of melt pool width, length, or cooling rates. Physics-based simulations can provide this information, which allows for more insight into the processing conditions.

Table 4.

AM Benchmark 2018 single track laser scan results on EOS 270 machine [20]. The expanded uncertainty, u, is normalized by the average depth to provide a percentage. Predictions within this percentage are within the estimated uncertainty of the measurements.

Case	Laser Power (W)	Scan speed (mm s−1))	D4σ (μm)	d (μm)	d/2σ	u/d (%)
A	150	400	100	150.75	3.0	12.6
B	195	800	100	91.28	1.8	10.1
C	195	1200	100	60.54	1.2	10.1

Table 5.

Comparison of AM Benchmark 2018 simulation results [26] and scaling law predictions for melt pool depth. Values are the percent difference between the predicted and measured values (i.e., difference divided by the average). Simulation results provided by the AM Benchmark committee.

Case	M1	M2	M3	M4	M5	M6	M7	Rubenchik et al. (2018) [10]	Fabbro (2019) [12]	Ye et al. (2019) [11]	Adjusted Ye et al. [11]
A	−67.1	−84.0	−83.4	−68.3	−61.3	−120.0	−54.7	−13.9	−41.2	−51.2	−2.6
B	−58.4	−57.8	−51.3	−49.6	4.0	−95.2	−25.3	−6.8	−22.5	−68.5	−1.0
C	−53.5	−33.0	−36.2	−29.4	−19.1	−75.7	−6.0	−6.3	−17.7	−92.6	−11.8

7. Conclusions

The effect of varying laser spot size was characterized for single scan tracks on IN625 and analyzed using scaling laws. These scaling laws enable process parameters for a known material and laser characteristics (laser spot size) to be scaled to other materials, parameters, and laser characteristics for quick engineering decisions. This study further investigated how well these scaling laws work for changing laser spot size and evaluated the ability of such scaling laws based on their melt pool depth predictions. The conclusions are as follows:

1. Laser spot size has a systematic effect on the solidified melt pool shape. Due to the nature of keyholing or laser drilling, melt pool depth was highly sensitive to laser spot size. In addition to laser power and scan speed, the laser spot size should be considered a critical process parameter with potentially large effect on melt pool shape and cooling rate.

2. All three scaling laws showed a linear response for their respective dimensionless parameters. Further investigation showed that the scaling laws predict melt pool depths around a 50 % difference from measured depths with some cases having a difference as high as 100 %.

3. Melt pool depth predictions for all three scaling laws are highly dependent on the absorption coefficient. Incorporating a non-constant absorption can improve the scaling law accuracy (< 35 % difference) for melt pool depths provided that the absorption dependence from conduction through keyholing mode is accurately known.

4. A comparison of scaling law predictions and thermal simulations for three cases in the transition and keyhole mode show that the scaling laws are comparable or better than some numerical simulations of the LPBF process. This is because the scaling laws contain some empirical constants that have been determined for a wide range of conditions; whereas, thermal simulations often lack some physics to accurately predict keyholing.

Highlights:

• Effect of laser spot size on melt pool morphology was systematically studied.

• Scaling laws provide a basis to compare the effect of laser spot size on melt pool depth.

• Absorption coefficients are critical for accurate melt pool depth predictions.

• Melt pool depth predictions from scaling laws show comparable accuracy to more complex simulation predictions.

Appendix A: Universal functions from Rubenchik et al. (2018) [10]

Rubenchik et al. [10] performed a regression analysis on simulation results by fitting the melt pool depth, length, and width as a function of normalized enthalpy, Peclet number, and laser spot size. Rubenchik et al. [10] defines B and p as

$$B=\frac{\mathrm{\Delta }H}{2^{\frac{3}{4}}\pi h_s}=\frac{AP}{2^{\frac{3}{4}}\pi h_s\sqrt{\pi Du{\sigma }^3}},\text{and}$$ (16)

$$\text{and}p=\frac{D}{\sqrt{2}\sigma u}\text{.}$$ (17)

The normalized melt pool depth, length, and width are given by:

$$d(B,p)=\frac{\sqrt{2}\sigma }{\sqrt{p}}\left[\begin{array}{c}0.008-0.0048B-0.047p-0.099Bp\\ +(0.32+0.015B)p\ln B\\ +\ln B(0.0056-0.89p+0.29p\ln p)\end{array}\right],$$ (18)

$$l(B,p)=\frac{\sqrt{2}\sigma }{p^2}\left[\begin{array}{c}0.0053-0.21p+1.3p^2+(-0.11-0.17B)p^2\ln p\\ +B(-0.0062+0.23p+0.75p^2)\end{array}\right],$$ (19)

$$w(B,p)=\frac{\sqrt{2}\sigma }{B{p}^3}\left[\begin{array}{c}0.0021-0.047p+0.34p^2-1.9p^3-0.33p^4\\ +B(0.00066-0.0070p-0.00059p^2+2.8p^3-0.12p^4)\\ +B^2(-0.00070+0.015p-0.21p^2+0.59p^3-0.023p^4)\\ +B^3(0.00001-0.00022p+0.0020p^2-0.0085p^3\\ +0.0014p^4)\end{array}\right]$$ (20)

Figure A1 shows a comparison of the predicted melt pool dimensions from the above equations to those measured in this paper. There is rather poor agreement for the range of experiments. Note it is not readily apparent if the thermophysical properties used in this study are different from Rubenchik et al. [10]; however, we do not believe any difference is the cause of the poor agreement. It is likely that that the thermal simulations that were used to develop the universal functions are missing some important physics for keyholing.

Figure A1.

Rubenchik et al. [10] predicted melt pool dimensions using the universal equations above versus the measured values.

Appendix B: Comparison of melt pool aspect ratio definitions

Figure B1.

A comparison of the melt pool depth, d, over the spot radius, 2σ, and the melt pool depth over the melt pool half width, 0. 5w. These ratios are nearly identical at the transition from conduction to keyholing mode (at a value equal to 1) shown with the vertical and horizontal lines. The diagonal line has a slope of 1 indicating that the ratios significantly deviate from each other at values > 1.

Appendix C: Representative micrographs for AMMT machine tracks

Figure C1.

Cross-sections of single track laser scans on a bare substrate produced on the AMMT machine for a fixed laser power and scan speed of 195 W and 800 mm s⁻¹ with increasing spot diameter from (a) to (g).

Appendix D: Representative micrographs for different Z offsets

Figure D1.

Cross-sections of single track laser scans for a fixed laser power and scan speed of 195 W, 800 mm s⁻¹ with different Z offsets.

Figure D2.

(a) Melt pool depth versus the build plate position (Z offset from the recoating plane) for three different spot diameters fixed at the recoating plane, (b) melt pool depth versus the estimated spot diameter accounting for the effect of the Z offset.