A dimensionless number for high-throughput design of multi-principal element alloys in directed energy deposition

Multi-principal element alloys (MPEAs) exist in large compositional spaces. With a potential compositional space of 5–35 at. % for each element, the number of possible alloy variations can include approximately 1 × 10⁶ distinct alloys for a five-component system.^1–4 Designing alloys in this large compositional space is challenging. Thermodynamic calculations can provide insight into equilibrium phase boundaries,^5–8 but available databases can limit the predictions. Thus, high-throughput (HT) methods to fabricate samples are required to characterize and validate predictions as well as test properties and build databases in the MPEA compositional space.

The HT fabrication methods of bulk, condensed inorganic samples at elevated temperatures are arguably in its infancy. Although HT or combinatorial methods exist and can facilitate testing and database development, bulk sample fabrication methods have been limited. Recently, directed energy deposition (DED) has been demonstrated as a HT fabrication method for bulk alloy production.^9–11 This process consists of varying the flow rate of powders from up to four hoppers, where the powders blend in a carrier gas and are distributed from a nozzle (Fig. 1). The stream of powders intersects with a laser, permitting in situ alloying. With calibrated flow rates, compositional changes can be made, and 25 compositions of roughly 1 cm³ specimens can be fabricated within a 5-h period. However, the challenge of this HT manufacturing is the ability to predict the process conditions without trial and error.

Laser-based additive manufacturing (AM) techniques, in general, have high cooling rates. The small focal point of the moving laser (typically less than 1 mm) and the associated small molten volumes on a larger cooled heat sink provide high cooling rates from 10² to 10⁶ K/s with DED methods having cooling rates in the lower half of this range.^12–15 For alloys, and in particular, MPEA sample fabrication, the high cooling rates associated with the DED process help reduce microsegregation with the potential to exhibit non-equilibrium solidification and retention of metastable phases. MPEAs are often characterized as exhibiting sluggish diffusion,^16,17 so the reduced scale of dendritic microsegregation will either decrease required homogenization times or enable metastability for enhanced properties or functionality.^18–21 

This study proposes a design strategy for the HT fabrication of MPEAs. Using the proposed methodology, unknown process parameters can be defined for a new material. Moreover, tailoring the cooling rate to increase the metastability is included to facilitate MPEAs' fabrication and increase the potential of metastability in the fabricated samples. Linking two length scales (macro- and meso-scale) offers a useful method in the design process. A dimensionless number is proposed to provide a mass and energy balance approximation to predict the processing of alloys in the DED process.

While dimensionless numbers are widely used in fluid mechanics, their use in additive manufacturing (AM) is still under development. A dimensionless number is expressed by a group of parameters, related to the process as well as inherent to the material, and could be effectively utilized to study additive manufacturing processes.^22–24 The advantages of using a dimensionless number in additive manufacturing techniques can be found elsewhere.^24,25 In this study, a dimensionless number has been developed for directed energy deposition (DED) as a function of process parameters such as laser power, laser spot size, feed rate (laser traverse speed), layer thickness, hatch spacing, and mass flow rate as well as material properties such as thermal diffusivity, specific heat capacity, latent heat of fusion, and melting point. To accomplish this goal, the Buckingham PI theorem²⁶ has been applied to derive a relationship among processes as well as machine parameters and properties of interest. Assuming physical and geometric properties are a strong function of the global energy density, $Eg$⁠, the dimensionless number, Π, can be written in the following form (the definitions and form of the terms in the equation are given in Table I):

where P, v, D_l, α, k, ρ, C_p, H, H_f, T_m, T_o, and $ṁ$ are laser power, feed rate, laser spot diameter, thermal diffusivity, thermal conductivity, density, specific heat capacity, latent heat, latent heat of fusion, melting temperature, ambient temperature, and mass flow rate, respectively. Global energy density, E_g, can be defined as the ratio of the laser power to the feed rate times laser spot size²⁷ as shown in Table I.

Using the Buckingham PI theorem in Eq. (1), a dimensionless number can be derived as follows:

In this study, four variables (i = 4) namely, $Eg$⁠, $α$⁠, H, and $ṁ$⁠, were chosen, and three dimensions were involved (M, L, and T) (j = 3). Thus, according to the Buckingham PI theorem,²⁶ at least one dimensionless number, Π ≤ i − j = 4 − 3 = 1 can be developed. The dimensionless number expressed in Eq. (2) represents the ratio of the input laser energy to melt the material with respect to heat dissipation. However, in a DED system, the choice of the layer thickness and hatch spacing is dictated by the laser power and laser spot size. In addition, properties, such as density and build height, are also influenced by layer thickness (Z) and hatch spacing (h). The melt pool dimensions are dictated by the laser power, feed rate, and laser spot size. At a constant laser power and feed rate, if the hatch spacing is larger than the melt pool width, then overlap of the melted regions will be incomplete and a poor build. If the layer thickness is higher than the melt pool depth, delamination will take place between the layers. Hence, a relationship is required to avoid these types of defects in the printed components. Thus, at a constant energy density, the layer thickness and hatch spacing could not be arbitrary chosen. To accommodate this effect in this study, a dimensionless length (L^* = Z/h), which is a ratio of the layer thickness to the hatch spacing, was introduced into Eq. (2). The addition of the dimensionless length is expressed in Eq. (3) as follows:

Based upon Rosenthal temperature distribution simulation,²⁸ the laser spot size can be correlated with layer thickness and hatch spacing. A range of values for layer thickness and hatch spacing are recommended as follows:

The dimensionless number described in Eq. (3) can be applied for both pre-alloyed powders as well as in situ alloying of multi-component alloys such as MPEAs. In general, the number is a mass and energy balance of the processing conditions and thermophysical properties of the materials. In the case of all alloys, properties can be evaluated by taking averages as described as

where S is the property of interest used in Eq. (5) and $xi$ and $Ni$ are the atom percentage and property of interest of the i-th element, respectively. The dimensionless number demonstrated in Eq. (3) was used to develop relationships among measured physical properties as well as physical dimensions. Specifically, the dimensionless number was used to express the physical dimension of build height. To demonstrate the application of this concept, 316L stainless steel (SS) and a range of compositions within the MPEA system of Cr, Fe, Ni, and Mo were processed. Material properties²⁹ are listed in Table II.

The DED machine used in this study was an Optomec LENS MR-7. Spherical, gas-atomized 316L stainless steel powder in the size range 45–150 μm (purchased from Stanford Advanced Materials) was filled in one of the four hoppers connected to the LENS MR-7. Ar flow gas brought the powder from the hoppers into the path of the 1 kW Nd:YAG 1070 nm wavelength laser with a 600 μm spot size and melted and deposited on the 316L SS build plate. The mass flow rate of the powder was controlled by changing the RPM (revolutions per minute) of the auger attached to the hopper. As defined above, in additive manufacturing, the melt pool geometry dictates the required hatch spacing and layer thickness. In the DED process, the melt pool is defined by the power, velocity, and powder feed rate.³⁰ Thus, the following process parameters were varied to analyze the variation of build heights—laser power (P), feed rate (v), powder mass flow rate (⁠$ṁ$⁠), laser hatch spacing (h), and layer thickness (Z). The scan strategy was kept constant for all samples—a bi-directional, single pass of laser beams for each print layer and a 90° rotation of scan vectors for each subsequent layer. Although the scan strategy will affect the heat transfer and, therefore, affect the melt pool geometry, defining a stable build at a constant scan strategy is a required first step. Each sample was printed for ten layers with each print layer being square shaped with dimensions 6.35 × 6.35 mm². During printing, the stand-off distance (i.e., the distance between the nozzle tip and the build plate) was set to 9.53 mm. The range of the process parameters used in this study is described in Table III. For 316L SS, laser powers were varied from 200 to 600 W in increments of 100 W, the layer heights were varied from 0.25 to 1.25 mm in increments of 0.25 mm, hatch spacings were varied from 0.25 to 0.51 mm in increments of 0.13 mm, and two laser feed rates of 6.35 and 10.58 mm/s were used. Each sample represented a unique process condition. Most 316L samples with a high heat input (Π₂ > 80) were rejected. Some of the process parameter combinations with high heat inputs (Π₂ > 80) had severe volatilization, and many of these prints had to be interrupted due to safety concerns. Because these samples were not printed for all ten print layers, the build heights could not be compared to the other samples and, hence, were rejected. For some other process parameters with high heat inputs (Π₂ > 80), the build had a non-uniform build shape, making the measurement of build height inaccurate and not reportable. Finally, some low heat input samples (Π₂ < 20) displayed irregular build morphologies, apparently owing to insufficient melting. Most of these were rejected samples.

316L SS was chosen as the model system to first demonstrate the effectiveness of the dimensionless number. The units for global energy density, thermal diffusivity, mass flow rate, and enthalpy [Eq. (3)] were used as J/mm², mm²/s, g/s, and J/g, respectively, to calculate the dimensionless number values described in Eqs. (2) and (3). A normalized build height vs dimensionless number relationship is shown in Fig. 2(a) for 316L SS. A normalized build height was used as a convenient way to represent the build height, as it is the ratio of the printed sample height to the set height per unit powder flow rate. This normalized build height can be expressed as follows:

where $h*, hactutal, n, Z, and ṁ$ are the normalized build height, actual build height, number of layers printed, layer thickness, and mass flow rate, respectively. The actual build height is the height physically achieved from the deposition of a prescribed number of layers, whereas the number of layers times layer thickness indicates the set height at the beginning of the print. In Fig. 2(a), the build height increases with the dimensionless number, and this can be attributed to the increased amount of input power. In addition, a linear relationship between the normalized build height and the dimensionless number has been fitted from the experimental observation and can be expressed as

Following the high-throughput (HT) analysis of build height, a few samples of the 316L SS were selected for low-throughput (LT) microstructural analysis. Since LT characterization is considerably more time-consuming, only a few samples at different dimensionless numbers were chosen to investigate the primary dendrite arm spacing (PDAS). Detailed studies of microstructural variation along the build direction in AM 316L samples have been previously reported.³¹ In this work, the PDAS was measured halfway along the build height of each sample in the center of melt pools. The plot in Fig. 2(b) depicts the relationship between PDAS and the build height used for the 316L SS samples. Most of the samples exhibited only primary dendrites. The cooling rate during DED printing of these samples was high enough to suppress secondary dendrites from growing in most of the samples. Although PDAS is not a rigorous estimate of the cooling rate in samples (compared to secondary dendrite arm spacing),^12,32,33 it is often used in the literature as a gauge of thermal history.^13,34 Nonetheless, the change in dimensions in the meso-scale is apparent. The DED-manufactured samples in the current study [Fig. 2(b)] exhibit primary dendritic arm spacing (PDAS) in the range 3.3 ± 0.3–8.7 ± 0.6 μm. This denotes a significantly slower cooling rate compared to the laser powder bed fusion (LPBF) manufactured 316L SS³⁴ but still fast enough to retard the growth of the secondary dendrites in most of the samples. The dendrite arm spacing values are also consistent with those reported in other studies,^12,14,15 where observed dendrite spacings of ∼3 μm estimated the cooling rate in DED to be around 10³–10⁵ K/s. For 316L SS, the relationship between the PDAS (l₁) and cooling rate (e) is given by l₁ = 80e^−0.33.¹³ 

The significance of the use of the dimensionless number is twofold. First, the dimensionless number provides a predictive means for the macro-scale build height, eliminating trial-and-error methods. In the DED process, the mass flow rate coupled with the process parameters dictates a specific layer height. For an effective build, the computer-controlled layer thickness increase of the working distance between the sample and laser/powder focal point must match the layer build height. If not, the laser focus will not be constant, resulting in an undesirable build. Previously this relationship was defined through trial-and-error methods, whereas the proposed dimensionless number minimizes the time for process parameter optimization with a predictive design tool for a calibrated system. In addition to the macro-scale build height, the dimensionless number permits an estimate of the microsegregation length scale and possibly cooling rate. The build height of a sample reflects the total mass and energy required for each layer in the build process. Thicker layers (and consequently higher build heights for a constant number of the layer thickness increase) have lower cooling rates (and coarser microsegregation). Although further studies can help to refine that relationship, the predictive nature of the dimensionless number permits tailored cooling rates for an increased potential of metastability in DED processed materials.

After using the dimensionless number in predicting the DED-based AM of 316L SS, the methodology was extended to a Fe–Ni–Cr–Mo MPEA system. For 3D printing, individual elemental powders of Cr, Fe, Mo, and Ni in the size range 45–150 μm (purchased from American Elements) were filled in each of the four separate powder hoppers. The main difference in this application is the fact that for 316L, the composition did not change. For the MPEA, a range of compositions was investigated, which introduced some challenges. The individual elements in the powder mixture had quite different melting temperatures. Mo has the highest melting temperature of 2623 °C, while Cr has the lowest boiling temperature of 2672 °C. Furthermore, Cr has a very high vapor pressure and tended to volatilize easily. Therefore, it became a challenge to melt Mo without boiling away Cr. In addition to depleting Cr in the final build, the severe volatilization also led to surface roughness and porosities. This was tackled by using relatively low Π₂ values (which was done by keeping lower powers) during the printing step and incorporating a remelting laser pass after each print layer. This involved running the laser over the deposited layer without having any powders blowing. This step helped to melt any unmelted Mo in the final build. The powers used varied from 400 to 750 W depending on the compositions—50 W increase for every 5 at. % increase in Mo. The parameters used to synthesize the MPEAs are listed in Table III. The composition of the powder blend coming out of the nozzle and into the laser path was dictated by the independent control of the auger RPMs on each hopper. This immediate ability to change powder compositions facilitates in situ alloying in a high-throughput (HT) fashion (see the schematic in Fig. 1). The feed rate and hatch spacing were kept constant at 8.47 mm/s (20 in./min) and 0.38 mm (0.015 in.), respectively.

A relationship between the normalized build height of MPEAs and the dimensionless number is shown in Fig. 3(a). The linear fit in Fig. 3(a) can be expressed as follows:

This linear fit indicates that the normalized build height for the MPEAs is the same as for 316L SS [in Eq. (6)], which supports that the normalized build height of MPEAs can be predicted using the proposed dimensionless number. By following the linear fit in Eq. (8), the unknown process parameters of a material can be predicted. The details of the determination of unknown process parameters are explained in the supplementary material, Sec. S1.

The actual vs intended composition of the MPEAs is shown in Fig. 3(b). With a first level iteration, the compositions of the elements were within ±10% as determined with x-ray fluorescence (XRF). The XRF scans were performed using an X-200 XRF analyzer, a handheld instrument using a 40 kV, Rh anode source that enabled rapid composition measurement at the rate of 1 min/sample. A spot size of 3 mm diameter ensured that the measured compositions were averaged over a large enough sample surface area. The accuracy of the XRF measurements was calibrated using a 316L SS specimen, whose actual compositions were measured by Luvak, Inc. (The comparison is given in the supplementary material, Table SII.) The XRF-measured compositions were accurate to be within ±0.3 wt. %. Typically, errors in XRF measurements begin to increase for elements with atomic weights lower than aluminum. Since all the elements in the current alloy systems consisted of mainly transition elements or higher, XRF was a good high-throughput option to measure compositions. As can be seen, the printed alloy compositions span a significant portion of the compositional space—Fe varied 2–85 at. %, Ni 0–100 at. %, Cr 0–24 at. %, and Mo 0–30 at. %.

The dimensionless number proposed in this effort was effective in predicting the build height for both pre-alloyed 316L SS powder as well as for in situ alloying of elemental powder of an MPEA system. Although both systems contained the same constituent elements, this relationship did fit over a wide range of compositions with regression fits of R² = 0.96. The comparison of all the data is shown in Fig. 4.

A Zeiss LEO 1530 scanning electron microscope (SEM) was used for imaging and energy-dispersive spectroscopy (EDS). The phases present in all the alloy samples were measured via automated x-ray diffraction (XRD) in a Bruker D8 Discovery x-ray diffractometer (data provided in the supplementary material, Sec. S3). The micrographs in Figs. 5(a) and 5(b) show the microstructure of a single-phase FCC alloy of nominal composition Fe₆₀Ni₂₀Cr₁₀Mo₁₀ at two different magnifications, respectively. This MPEA was manufactured at a Π₂ value of 28 and built up to a height of ∼3 mm. The microstructure consisted of only primary dendrites with no evidence of secondary dendrite formation. The PDAS in this alloy was quite fine and was measured to be 3.4 ± 0.4 μm, indicating a cooling rate of ∼10⁴ K/s based on the equation for 316L stainless steel cited previously. EDS area maps [Figs. 5(c) and 5(d)] of the sample surface corresponding to Fig. 5(b) depict the elemental distributions of Fe and Mo, respectively. The Mo segregated into the interdendritic regions, while the Fe was depleted in these regions. Cr and Ni were distributed more uniformly throughout the microstructure. The Π₂ defined the parameters that controlled the cooling rate resulting in fine PDAS, and hence, a fine scale of microsegregation.

The normalized build height as a function of the dimensionless number should be further investigated with different alloy systems. Furthermore, detailed investigations on the dendrite spacing as a function of the build heights are required. Nonetheless, the dimensionless number, based upon a mass and energy balance of the thermophysical properties with respect to the process parameters, permitted a reliable prediction of processing conditions for macro-scale dimensional control. Meso-scale features, such as dendrite arm spacing and microsegregation spacing, can also be tailored with the term. The control of the dendrite arm spacing indicates the ability to tailor cooling rates that cover two orders of magnitude in the DED process and, thus, provides a design parameter for promoting increased metastability. The dendrite arm spacings are typically five times finer than an arc-cast specimen, and the shorter distances of microsegregation allow shorter homogenization times during postprocessing. Finally, the ability to generate HT bulk samples within a large compositional space and predict the appropriate process conditions for build effectiveness and optimized cooling rate permits useful design capabilities for HT sample fabrication. Specifically, the ability to generate bulk samples with different compositions, validate or refine thermodynamic databases, and provide samples for subsequent characterization and property assessments will enable more discovery in MPEAs.

See the supplementary material for the unknown process parameter development, XRF accuracy, and XRD data.

This work was partially supported by the Advanced Research Projects Agency-Energy (ARPA-E) ULTIMATE program (Grant No. DE-AR-0001431), ARPA-E (Grant No. DE-AR-0001050), and National Science Foundation (NSF) Designing Materials to Revolutionize and Engineer our Future (DMREF) program (Grant No. 1728933) for the DED powder hoppers.