Numerical Microstructure Prediction for Lattice Structures Manufactured by Electron Beam Powder Bed Fusion

Abstract

The latest advances in additive manufacturing have given rise to an increasing interest in additively built lattice structures due to their superior properties compared to foams and honeycombs. The foundation of these superior properties is a tailored microstructure, which is difficult to achieve in additive manufacturing because of the variety of process parameters influencing the quality of the final part. This work presents the numerical prediction of the resulting grain structure of a lattice structure additively built by electron beam powder bed fusion. A thermal finite-difference model is coupled to a sophisticated cellular automaton-based crystal growth model, including nucleation. Numerically predicted grain structures, considering different nucleation conditions, are compared with experimentally derived EBSD measurements. The comparison reveals that nucleation is important, especially in fine lattice structures. The developed software, utilizing the nucleation model, is finally able to predict the as-built grain structure in lattice structures.

1. Introduction

Energy conservation and weight reduction are becoming increasingly important in today’s industry. The main approach to addressing these challenges lies in the application of lightweight materials to numerically optimized structures [1]. These optimized structures may be stochastically derived, like foams, or deterministically constructed, like honeycombs or lattice structures. While foams and honeycombs are nowhere near the optimum for many applications, they dominated the field of artificial cellular structures for a long period of time due to their ease of producibility [2]. The latest advances in additive manufacturing (AM) gave rise to an increasing interest in additively built lattice structures. Lattices outperform both foams and honeycombs in terms of being lightweight, high-strength, absorbing energy, and reducing vibration [3]. Saving material, time, and energy during the manufacturing process [4], as well as optimizing the strength and simultaneously reducing the weight of the resulting part [5], are the main driving forces behind this growing significance.

The determination of the suitable manufacturing process and material for the fabrication of lattice structures has to consider several aspects, including size constraints, morphological uncertainty, and self-supporting properties [6]. Due to the complexity of the structures, lattices are especially prone to manufacturing defects [7]. Special design rules for additively manufactured lattice structures have been identified, starting from the unit cell design until verification and optimization [8]. Tang et al. [9] proposed a design and optimization strategy for lattice structures with consideration of additive manufacturability. With optimized process parameters and complying with the design rules for AM, highly filigree structures are feasible, mimicking their biological counterparts [10].

From the variety of metal-based AM processes, powder bed fusion (PBF) processes are among the most mature manufacturing processes for the fabrication of lattice structures [11]. PBF processes share the common principle of building the resulting part in layers by selectively melting distinct areas of a powder bed. Many researchers have investigated the mechanical performance of lattice structures manufactured by PBF. Matheson et al. [12] produced stochastic open-cell aluminum foam samples conventionally by investment casting and subsequently manufactured two copies of the resulting foam additively using laser powder bed fusion (PBF-LB). Both replicas exhibited distinctively differing failure events, indicating variability among parts despite identical production parameters. Apart from this phenomenon, the additively manufactured samples showed an average grain size that was an order of magnitude smaller than that of the original sample. Traxel et al. [13] investigated the compressive performance of five different unit cell designs built additively from Ti6Al4V powder using PBF-LB. Under compressive loading, significant differences in both the compressive strength and the elastic modulus were found.

The influence of the build orientation as well as the microstructure on the mechanical properties of Ti6Al4V samples manufactured by laser metal wire deposition was investigated by Åkerfeldt et al. [14]. Specimens oriented parallel to the deposition direction showed higher yield strength, while perpendicular-oriented samples exhibited significantly higher tensile elongation. Delroisse et al. [15] studied the impact of the inclination angle on single struts fabricated from AlSi10Mg processed by PBF-LB. Weißman et al. [16] followed a similar approach, albeit with consideration of several struts fabricated from Ti6Al4V via PBF-LB as well as PBF-EB. They found significant differences in the microstructure of the resulting samples that are reflected in their mechanical properties. Jeong et al. [17] investigated the influence of the beam continuity in PBF-EB on the microstructure and mechanical properties of lattice structures manufactured from commercially pure Ti powder. The differing heat input due to varying scan strategies resulted in significant differences in the coarsening of the emerging $\alpha$-Ti grains. Biffi et al. [18] compared the resulting microstructure of lattice structures and bulk material manufactured by PBF-LB from TiNi powder. Beside the emergence of different phases, the grains in the lattice structure were oriented in preferred directions in contrast to those in the bulk material.

While the correlation between microstructure and mechanical properties is frequently observed, modeling the mechanical properties of lattice structures focuses mainly on geometrical considerations. Originating from the pioneering work of Gibson and Ashby [19], homogenization and FE models are applied primarily nowadays [20,21]. De Pasquale et al. [22] studied the mechanical properties of Ti6Al4V lattice samples built by both PBF-LB and PBF-EB and compared the results with a numerical model based on the homogenization method. The differing process conditions resulted in different lattice geometries of the samples and, accordingly, diverging mechanical properties. Munford et al. [23] developed a model to predict the anisotropic apparent modulus and strength of stochastic and rhombic dodecahedron structures based on lattice density and fabric tensors. Only a few authors are incorporating microstructural modeling into their investigations concerning lattice structures. Dong et al. [24] applied the commercially available software Fluent to calculate the flow, heat, and mass transfer of AlSi10Mg during PBF-LB for an inclined lattice strut. The simulation results were used to explain the microstructural heterogeneities observed experimentally in different zones of the lattice by thermal means. Johnson et al. [25] applied a kinetic Monte Carlo model to simulate the resulting microstructure of a 304L stainless steel cylinder manufactured by the Laser-Engineered Net Shape process. While the simulation results coincided with the experiments regarding the transition from equiaxed to columnar grains, the numerically derived grain size exceeded the measured value significantly.

Measurements on differently oriented samples manufactured additively [26] as well as first principle calculations [27] show the mechanical properties of nickel-base superalloys are strongly dependent on the direction of load in relation to the crystallographic texture. This behavior is of special interest in combination with lattice structures due to the distinct tilting of the struts. Kavousi et al. [28] showed a combination of geometrical and crystal plasticity modeling for AM intended for applications with small struts. The simulation domain in this case only covered a few beam paths. Tan and Spear introduced a Multiphysics modeling framework to predict the Process-Microstructure-Property relationship in PBF, albeit only for bulk material without consideration of the shell region of small struts. Li et. al. [29] used a cellular automata (CA) approach on two distinct layers to estimate the impact of process parameter changes on the physical properties of a vehicle chassis fabricated by wire and arc additive manufacturing. Chen et. al. [30] applied a CA approach to estimate the effects of process parameters on the microstructure of Inconel 718 during powder bed fusion-based. Their work concentrated on a domain of a few hundred micrometers and disregarded boundary effects.

In a previous work [31], Koepf et al. predicted numerically the resulting microstructure for a vertical strut manufactured additively by PBF-EB from CMSX-4 powder and validated the simulation with experiments. The cylindrical structure exhibited a distinct shell of randomly oriented grains growing from the surrounding powder bed, surrounding a core of columnar grains with their <001>-growth direction aligned with the build direction.

Based on this research, the present publication investigates the as-built microstructure of a lattice structure’s elementary cell manufactured by PBF-EB from CMSX-4 powder. The thermal conditions responsible for the developing microstructure are studied for different layers exhibiting distinct scan profiles. The numerically predicted microstructure is validated by means of electron backscatter diffraction (EBSD) measurements. This contribution aims to improve the understanding of the relationship between process parameters and microstructure within complex geometries. It facilitates the fabrication of tailored parts, for example, by manipulating the shell zone or deliberately inclining grains.

2. Materials and Methods

2.1. Numerical Models

The numerical prediction of an additively manufactured microstructure comprises at least a thermal and a grain structure model. The thermal model determines the current solidification front of the melt pool. It is coupled to a grain structure model, which covers grain growth, grain selection, and nucleation, to predict the as-built microstructure.

Previous works [31,32] used the in-house developed software SAMPLE3D for the numerical prediction of the as-built microstructure of additively manufactured parts. In these contributions, the investigated geometries were rotational-symmetric, enabling the reuse of a single thermal field calculated up front. With the constantly changing melting profile exhibited by the lattice structures investigated in this work, this simplification is no longer feasible. Instead, the resulting thermal field for each layer must be calculated individually, significantly increasing the computational load. Consequently, the fast and reliable computation of the thermal field induced by the traversing beam is paramount for an efficient prediction of the as-built microstructure. A finite difference (FD) approach was selected due to its fast execution speed, straightforward implementation, and excellent parallelization capability. Furthermore, it harmonizes well with the cubic cells applied in the cellular automaton (CA) approach we are using for calculating crystal growth. This CA model, in turn, was selected due to its excellent results without introducing a lattice dependency [31,32].

The applied FD approach is introduced in the following subsection. Subsequently, the CA model, including a nucleation model [33] for the microstructure prediction as well as the loose coupling between thermal and microstructure calculations, is reviewed.

2.1.1. Finite Difference Method

The foundation for the numerical prediction of the thermal field in PBF is given by the non-linear heat equation [34]

$$c_p\frac{\partial T}{\partial t}\equiv \frac{\partial H\left(T\right)}{\partial t}=\frac{\lambda }{\rho }\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+\frac{\dot{Q}}{\rho },$$

(1)

where $T$ denotes the temperature, $H$ the enthalpy, $t$ the time,$\rho$ the density, $c_p$ the specific heat capacity, and $\lambda$ the constant thermal conductivity.

The source term $\dot{Q}$ represents the heat incorporated into the material on the x-y-plane from the traversing beam. Following the work of Sanders [35], it is modeled by a two-dimensional Gaussian distribution

$$\dot{Q}(x,y)=\eta ·P·\frac{8}{\pi ·d_B^2}\exp \left(-\frac{8}{d_B^2}\left({\left(x-x_B\right)}^2+{\left(y-y_B\right)}^2\right)\right),$$

(2)

where $P$ denotes the beam power, $d_B$ the $4\sigma$ beam diameter, and $x_B$ the current beam location. The efficiency factor $\eta$ distinguishes between a laser and an electron beam. Within this work, the efficiency factor $\eta$ was set to unity.

Equation (1) is discretized according to the FD method by applying a forward-time-central-space (FTCS) approach [36] on a Cartesian grid with a constant cell size in all dimensions. With $i,j,k$ denoting the indices of a distinct cell, the temperature for a given cell at the time $n+1$ is predictable from the values of the adjacent cells at the current time step $n$ according to

$$H_{i,j,k}^{n+1}=H_{i,j,k}^n+\frac{\lambda }{\rho }\frac{\mathsf{\Delta }t}{\left(\mathsf{\Delta }x\right)2}\left(T_{i-1,j,k}^n+T_{i+1,j,k}^n+T_{i,j-1,k}^n+T_{i,j+1,k}^n+T_{i,j,k-1}^n+T_{i,j,k+1}^n-6T_{i,j,k}^n\right)+\frac{\mathsf{\Delta }t}{\rho }·{\dot{Q}}_{i,j,k}^n,$$

(3)

where $∆t$ represents the time step size and $\mathsf{\Delta }x$ represents the cell size.

In addition to the heat input discussed above, flux boundary conditions (BC) are applied with a distinct heat flux defined up front (Neumann-BC) to all boundary faces. The resulting values at the boundary cells are calculated using a fictitious cell approach [36]. From preliminary works, the boundary heat flux was set in this work to 100 kW/m².

The actual temperatures are calculated from the enthalpy using tabulated material data showing the T-H-relationship [37].

FTCS is an explicit numerical approach, resulting in a coupling of the temporal and spatial time step sizes. It is stable if the von-Neumann-stability criterion for three dimensions is fulfilled [38]:

$$\frac{\lambda }{\rho c_p}\left(\frac{∆t}{\left(\mathsf{\Delta }x\right)²}\right)\le \frac{1}{6}$$

(4)

The timestep sizes used in this work were calculated up front to fulfill this stability criterion. A detailed illustration of the workflow is shown in Appendix A.

For now, no melt pool dynamics are considered. This topic will be addressed in future work.

2.1.2. Cellular Automaton Approach

The crystal growth is modeled by a CA—originally developed by Gandin and Rap-paz [39] and adapted for PBF processes [31,32]. In contrast to level set [40] or phase field methods [41], the CA approach does not represent the growing dendrites individually. Instead, it models the growth of complete grains on a Cartesian grid by tracking their envelope through the thermal field during solidification.

For each grain inside a cell, an octahedral hull exists, which is rotated according to the orientation of the main growth directions of the dendrite. All octahedrons neighboring the liquid melt pool can grow according to the undercooling prevailing at the center of the cell associated with it. A kinetics law, modeling the growth velocity independence of the local undercooling, is used [33]. As soon as the growing octahedron comprises the center of an adjacent liquid cell, this cell is “captured” and incorporated into the solidifying grain. During capture, a new octahedron mimicking the orientation of the parental grain is initiated and associated with the captured cell. The new octahedron is truncated and repositioned [39].

While the capturing algorithm during the solidification of casting processes is well documented in the literature, the remelting of previously consolidated material was originally not necessary because only directional solidification was modeled. The iterative melting and solidification of material during the PBF process necessitates consideration of this effect. An efficient solution was found for storing the initial size of the octahedron of each cell [35]. During the remelting of a distinct cell, the associated octahedron is reset to its initial size. Following this approach assures an adaptation of the grain envelopes to the dynamically changing melt pool without interference from the growth algorithm.

In combination with the capturing algorithm, a novel kinetic law for the crystal growth model was developed [33] by applying the phase field method. Analyzing the velocity-undercooling behavior using the commercially available software MICRESS (Version 6.4) [42] results in the following power law that is also used in this work for calculating the growth velocity $v$ in dependency of the undercooling $∆T$:

$$\left(∆T\right)=1\times {10}^{-6}\times {∆T}^{-2.8}$$

(5)

2.1.3. Nucleation Model

The nucleation model applied in this work was not introduced until recently [33] and is recapped here briefly. The development of the original crystal growth algorithm for stationary systems like casting simulations resulted in a nucleation model inapt for PBF processes. A simple but effective approach for consideration of nucleation effects was found in a supplementary application of the growth algorithm on nuclei seeded in front of the solid-liquid interface [33]. The newly formed nuclei are treated as normal grains, being subject to the same thermal conditions as the consolidation material, although growth occurs in a separate “shadow grid” detached from the regular grains. If the growing nucleus reaches a critical radius before the containing cell is captured by the advancing solid-liquid interface, it is incorporated into the regular grid as a newly emerged grain equitable to its established neighborhood. If the cell containing the nucleus is captured earlier, the seed is erased, with no nucleus emerging.

Based on observations reported by different researchers in the field of PBF, a heuristic approach is applied to consider new grains within the CA-based crystal growth model. A shadow grid, experiencing the same thermal field as the CA, is used to contain nucleation cells. Each nucleation cell contains a nucleus of random orientation, growing according to the same growth kinetics as the regular grid. When the size of a nucleus exceeds a defined critical radius (CR) before the nucleation cell is overgrown from the solidification front, it is incorporated into the regular grid, disturbing the approach of the original solidification front. Preliminary works showed reasonable values of CR to be in the magnitude of the primary dendrite arm spacing. In this work, CR is set equal to the cell length.

2.2. Material Model

The CA model shown in this work was applied previously to the nickel-based superalloys IN718 and CMSX4, albeit with differing heat models. The FD model used for calculating the thermal field in this work is validated using both materials.

Due to its combination of good weldability, forgeability, and strength at high temperatures, the nickel-base superalloy IN718 has been the workhorse for the aero-engine sector [43]. Its excellent weldability has made it a natural choice for early-stage additive manufacturing with LBB and EBB processes [44].

The development of superalloys specially designed for producing single-crystalline (SX) parts started in the 1970s with the idea of eliminating all high-angle grain boundaries from directionally solidified (DS) castings [45]. CMSX-4, belonging to the second generation of those alloys, contains 3 wt.% Re, resulting in a significant improvement in creep strength [46]. CMSX-4 is considered non-weldable due to the high shares of Ti and Al within the alloy, resulting in significant difficulties in processing the material additively. The production of crack-free parts from CMSX-4 using PBF-EB is feasible, as Ramsperger et al. [47] exemplarily showed.

The material values for CMSX-4 (shown in Figure 1a–c) as well as IN718 (shown in Figure 1d–f) were adapted from Mills [37] using linear interpolation. Within the mushy zone, multiple supporting points were considered depending on the fraction of solids. The distinct values between the supporting points are calculated by linear interpolation. Values are modeled beginning at room temperature (20 °C). In Figure 1, the focus is laid on the interval between 1000 °C and 1600 °C to show the mushy zone in detail.

3. Software Validation

The predictive capabilities of the crystal growth model applied in this investigation were demonstrated previously in combination with different heat sources, materials, and scanning strategies: The columnar texture within the bulk of PBF-EB-manufactured IN718 samples was impressively mimicked by applying an analytical heat source without nucleation [32]. A similar investigation with an adapted scan strategy and consideration of nucleation proved the impact of nucleating grains on the as-built microstructure [43]. In combination with a heat solver based on the Finite-Element-Method, the as-built microstructure of a thin cylinder was successfully replicated [31]. The grain structure of another Ni-based alloy, ABD^®-900AM, was successfully predicted using a similar FD approach with a volume heat source [48].

Prior to the application of the crystal model, the FD heat solution applied in this work was validated for different materials by comparing the numerically predicted melt pool with those derived from the analytical solution as well as experiments. For this investigation, CMSX-4 and IN718, two prominent nickel-base superalloys, are selected. Due to its combination of good weldability, forgeability, and strength at high temperatures, the IN718 is considered the workhorse in additive manufacturing [44]. CMSX-4, on the other hand, belonging to the second generation of superalloys specifically designed for single-crystalline parts, has become of interest for PBF more recently [48].

3.1. Single-Line Experiments

Single-line melting experiments were conducted by Hartmann [49] using an Arcam™ (GE Additive Arcam, Mölnlycke, Schweden) S12 machine. The lines were melted at room temperature in solid blocks of CMSX-4. The parameters applied for this investigation are summarized in

Table 1. Beam parameters applied for validation of the analytical and numerical heat solutions by comparison with single-line experiments.

Parameter	Value	Parameter	Value
Scan velocity	0.5–2.0 m/s	Line energy	0.15–2.4 J/mm
Beam power	300–1200 W	Preheat temperature	20 °C

. The specimens were metallographically prepared by longitudinal section cuts conducted perpendicular to the scan direction, and the melt pool depth was measured using optical microscopy. The experiments were mimicked using an analytical heat source [32] as well as the numerical solution introduced in this work. In both cases, a spatial resolution of 10 μm and a time step size of 2.5 μs were used. Melt lines of 10 mm were simulated using a computational domain of 300 × 1200 × 200 cells.

Figure 2 compares the resulting melt pool depths, determined using the liquidus temperature, with experimentally determined values. While the analytical solution overestimates the melt pool depth due to the lack of latent heat consideration, the numerically predicted melt pool depths are close to the experimentally determined values for all combinations of beam velocity and power.

3.2. Hatch Experiments

Hatching experiments by Helmer [50] using an ARCAM (AB GE Additive GE Additive Arcam, Mölnlycke, Schweden) A2 machine with IN718 powder were used for validation of the simulation results with overlapping heat fields. Cubic samples with an edge length of 15 mm were produced by applying a standard cross-snake hatching scheme. Details of the process parameters are listed in

Table 2. Beam parameters applied for validation of the analytical and numerical heat solution by comparison with layer hatching experiments with IN718.

Parameter	Value	Parameter	Value
Scan velocity	3.0–3.9 m/s	Scan length	15 mm
Beam power	510–780 W	Line offset	100 µm
Area Energy	1.7–2.0 J/mm2	Preheat temperature	950 °C

. The measured melt pool depth was again compared to the depth of the liquidus isotherm in the simulated results.

Applying the same spatial and temporal resolution as in the single-line validation results in a computational grid of 1500 × 1500 × 30 cells for the analytical calculation. For avoidance of heat piling up on the boundaries, a 3 mm buffer zone was used with the FD solution, resulting in a grid of 2100 × 2100 × 200 cells. Figure 3 compares melt pool depths predicted by analytical as well as numerical methods with those experimentally determined.

Like the single-line results, the analytical solution overestimates the melt pool depth significantly. The numerical results of the FD model are significantly closer to the experimentally derived values. Due to the powder consolidation, the measured melt pool depth does not represent the true heat penetration within the material. The powder bulk density affects the resulting thickness of the solidified material. The overestimation of the melt pool depth in the numerical solution results mainly from the negligence of these powder shrinkage effects. Assuming an average bulk density of 50% results in a doubling of the layer thickness.

4. Experimental and Numerical Procedure

4.1. Elementary Cell Design

The basis for this investigation forms a face-centered cubic elementary cell with an edge length of 4 mm, exhibiting both horizontal and vertically oriented struts (compare Figure 4). Autodesk™ (San Francisco, CA, USA) Inventor was applied for constructing the struts as well as their assembly, as shown in Figure 4a. The struts were constructed as cylinders with a diameter of 1 mm and an inclination angle of 45°.

The crossed struts in both horizontal and vertical directions result in constantly changing profiles along the build direction (shown exemplarily for three distinct planes in Figure 4a). At the bottom of the elementary cell, the crossing struts extend horizontally, resulting in a widespread area to be melted. Above this section, the vertical oriented struts exhibit significantly smaller melting areas. Figure 4b indicates the position of horizontal and vertical section cuts taken for validation. This work focuses on the microstructure of the vertical crossing struts. The complex thermal conditions within the horizontally oriented struts are discussed briefly.

4.2. Experimental Procedure

Multiple samples of the elementary cell were produced on an Arcam™ (AB GE Additive, Mölnlycke, Schweden) A2 machine localized at the Joint Institute of Advanced Materials and Processes (ZMP) in Fürth (Germany) using an acceleration voltage of 60 kV.

Electrode induction melting gas-atomized (EIGA) CMSX-4 powder with a size distribution between 45 µm and 105 µm was utilized. Although CMSX-4 is considered non-weldable, suitable process parameters enable the dense and crack-free production of filigree structures [31]. Based on this research, similar process parameters were selected in this investigation, albeit with a reduced beam power due to the small diameter of the struts.

Table 3. Process parameters for building the lattice structure.

Parameter	Value	Parameter	Value
Scan velocity	0.5 m/s	Layer height	50 µm
Beam power	200 W	Line offset	100 µm
Preheat temperature	1050 °C

lists the process parameters applied for the manufacturing of the elementary cell.

A cross-snake hatching scheme was applied, where the beam traverses the surface in a zigzag pattern. The beam direction rotates during the layers by 90 degrees, resulting in a repetition of the scanning direction every fourth layer. Iteratively alternating scanning directions and progressively changing melting profiles result in varying solidification conditions per layer.

Section cuts were conducted in the center of the horizontal as well as vertical-oriented struts, as indicated in Figure 4b. After grinding with SiC paper from P80 up to P4000, the samples were polished using 3 µm MOL as well as OP-chem polishing cloth by Struers for 7 min. Finally, the samples were etched for 5 s at 50 °C in a V2A stain. The resulting specimens were investigated by optical microscopy as well as EBSD measurements performed using a FEI Helios 600i dual-beam SEM located at the Joint Institute of Advanced Materials and Processes (ZMP) in Fürth, Germany.

4.3. Simulation Setup

To avoid heat accumulation at the borders, the domain for the numerical computation of the thermal field must be significantly larger than that required by the crystal growth model. In analogy to the approach shown in [31,33], the calculation of the thermal field and crystal growth are conducted separately. The distinct thermal fields for each layer were calculated up front and stored on the distributed storage device of the computer cluster. The beam paths for melting each layer were determined by slicing the geometry according to the layer height and applying a cross-snake scan pattern. The subsequent crystal growth simulation reads the corresponding thermal field for the distinct layer prior to the start of the crystal growth model. The simulation parameters for the thermal as well as the crystal growth simulation are listed in

Table 4. Simulation parameter used for thermal- and crystal growth simulation.

Parameter	Thermal	Crystal Growth
Domain	8 × 8 × 3 mm3	6.4 × 6.4 × 0.4 mm3
Cell size	10 µm	10 µm
Beam diameter	400 µm	-
Time step	2.5 µs	2.0 µs

. All other parameters were set in accordance with Table 3.

5. Results and Discussion

The thermal field at 0.5 mm build height is shown as an example of the lower half consisting of individual, colliding struts. Additionally, the layer at 2.5 mm build height is analyzed, showing the thermal conditions in the central region after the merging of both struts.

5.1. Thermal Conditions

Beside the thermal gradient and the cooling rate, the geometry and lifetime of the traversing melt pool are decisive for the resulting grain structure. Using the cross-snake hatching strategy, a persistent melt pool facilitates the development of columnar texture, with the preferred <001> growth direction being aligned with the build direction [31,32]. Individual melt lines not converging into a single, persistent melt pool, on the other hand, promote the development of nucleation and a variety of different grain orientations.

The resulting thermal conditions during the melting of the horizontally oriented struts are illustrated exemplarily in Figure 5 for the build height of 0.5 mm. The contour to be melted at this height is shown in the background. The color bar indicates the region from solidus to liquidus temperature. All temperatures below solidus temperature are not shown, and temperatures above liquidus temperature are colored red.

The upper part of Figure 5 shows the high-angle view of the surface of the complete domain for different time steps. This view clearly shows the melt pool for the distinct struts traversing persistently across the surface. The joining of the pools at the junction (Figure 5b) results in an enlargement of the melt pool and trailing melt pools towards the end (Figure 5c). The detailed view of cross-section cuts shown on the bottom part of Figure 5 shows no significant difference in the liquefied region between the beginning (a) and the end (c) of the layer.

The thermal conditions during the melting of the vertically oriented struts are shown exemplarily in Figure 6 for the build height of 2.5 mm. As above, the high-angle view is shown on top with the cross-section cut positioned below. These images clearly show the persistent melt pool developing in the vertically oriented struts as well.

5.2. As-Built Microstructure

Thermal calculations show the shape and stability of the developing melt pool and build the foundation for successive solidification simulations. The dendrites growing from the substrate are not aligned with the direction of the thermal gradient, though. A gap is developing between the solidification front and the liquidus isotherm, depending on the orientation of the preferred growth orientations of the individual crystals [39]. Without knowledge of the grain orientations of the grains growing from the substrate into the melt, this gap—and thus the undercooling prevailing at the dendrite tips—is not predictable from heat conduction calculations alone.

Insights into the solidification conditions resulting from distinct thermal conditions provide crystal growth simulations that consider the preferred growth orientation of the dendrites with a temperature-dependent kinetic growth model. The right-hand side of Figure 7 shows exemplarily the thermal gradient (b) and solidification velocity (c) resulting from the thermal field at four different build heights (a).

These depictions show the prevailing solidification conditions at the time of solidification, i.e., as they were captured by the advancing solidification front. Therefore, all cells that have been liquid at a specific point in time during melting and solidification are visible in this image, resulting in a colored area significantly larger than the size of the persistent melt pool shown above. The heat source in all these cases was set to scan the surface in a cross-snake pattern back and forth through the observation plane with a transversal movement from left to right.

Figure 7 shows the highest thermal gradient as well as the lowest solidification rate at the melt pool border, indicating a planar solidification front [51]. Under these conditions, new grains can only grow and prevail against already established ones when nucleating within the gap of two diverging grains and with a preferred growth direction better aligned with the thermal gradient.

To analyze this effect at different build heights, the crystal growth was calculated twice, once without consideration of nucleation as well as including the nucleation effect. Figure 8 compares both results (a and b, respectively) with experimentally derived EBSD measurements (c). Below each cross section, the corresponding pole figures are shown, taken from the central junction area of the structures (indicated by the white square).

While the shell region of the struts, consisting of grains not aligned with the build direction, is clearly visible in all cases, the pole figures, showing the preferred growth orientations of the grains in the central area, differ significantly. The experimentally derived EBSD shown in Figure 8c shows the main growth direction to be strongly aligned with the build direction. No preferred secondary growth direction is discernible. The simulation results with consideration of nucleation (Figure 8b) show a similar result, albeit with a less pronounced primary growth direction. The <110> growth direction at the borders seems to be more determinant than experimentally derived. The simulation result without consideration of nucleation (Figure 8a) shows a similar high amount of <110> grains prevailing at the borders of the strut. This phenomenon must be investigated further in future works. While there is also a preferred primary grain orientation developing in this case, it is significantly less pronounced than in the simulation result considering nucleation or the experimentally derived measurements. The differences between the simulation results with and without nucleation suggest nucleation plays a significant role in the development of the prevailing <100> grain orientation in the central region of the vertically oriented struts. This observation is further investigated by means of horizontal cross-section cuts through both the center of the horizontally oriented struts as well as the center of the junction between the vertically oriented struts.

Figure 9 shows a horizontal cross section cut at 0.5 mm build height (compare Figure 4). Again, the simulation without nucleation (a) and with consideration of nucleation (b) is compared with the experimentally derived EBSD measurement (c). In contrast to the results shown for the vertical section cut (shown in Figure 8), in this case, no significant difference in the primary grain orientation is discernible between the simulation with and without nucleation and the EBSD measurement. The shell region is dominated by randomly oriented grains.

The bulk in the center of the horizontally oriented struts consists of columnar, <100> oriented grains, as shown in the pole figures in the lower part of Figure 9. The results for the simulations with and without nucleation are almost identical. Well-oriented grains are developing due to grain selection, with no need for nucleation. Without displacement between different layers, no misoriented grains are growing into the bulk from the powder bed, and no gap is opening for the nucleation of new seeds.

A significant difference is discernible between simulation and experiment in the secondary growth direction, though: while the EBSD measurements again show no preferred secondary orientation in the experiment, the simulation exhibits a distinct secondary orientation aligned with the scan direction of the electron beam path. This effect of the simulation is already known from previous works [32], although its origin is still the subject of investigation.

Figure 10 shows horizontal cross-section cuts (top) and grain size measurements (bottom) taken at the central junction of the vertical struts (4th layer in Figure 7). The simulated grain structures with and without nucleation are shown in Figure 7a and Figure 7b, respectively. In Figure 7c, experimentally derived EBSD measurements are shown. The bottom of Figure 10 compares the distinct grain sizes measured by both intersection (d) and planimetric (e) measurements [52]. Thereby, the white lines indicate the lines used for intersection measurement, and the black lines mark the areas of planimetric grain size measurement.

As already seen in previous work [31], the planimetric measurement results in slightly larger grain sizes while not changing the trend. The grain size distributions in this case clearly show a constant grain size throughout the cross section. Only the grains growing into the part from the powder bed at the border show a smaller grain size that is in the range of the initial grain size within the powder. Comparing the grain sizes between the simulation results and the experiment reveals a significantly higher average grain size in the simulation, disregarding nucleation. With consideration of nucleation, the grain sizes mimic those experimentally derived. This result is another indication of the survival of new grains nucleating in the melt pool bottom region.

Preliminary works investigating the resulting grain structure of tilted struts using simulations without consideration of nucleation showed 100 and near-100 textures, regardless of the scanning strategy. The occurring nucleation seems to increase this effect, as shown by the analysis of the central junction of different struts in the current work. By taking grain structure-dependent mechanical properties into account in component optimization, the influence of the process strategy and control on the component properties could be analyzed. This approach will be used in future work to optimize process conditions and strategies for tailoring the resulting properties of additively built structures.

6. Summary and Conclusions

Based on a sophisticated crystal growth model, the grain structure of an elementary cell of a lattice structure was predicted and compared with experimentally derived EBSD measurements of PBF-EB-manufactured samples. The heat input of the electron beam was calculated up front for the individual layers with a FD heat solver specially developed to be applied in combination with the CA crystal growth model. The heat model was validated by comparison of the numerically predicted melt pool depths with experiments for single lines as well as hatching experiments. An alternative approach for modeling nucleation published only recently was applied to investigate the significance of nucleation for the resulting grain structure of a complex lattice structure. The following conclusions can be drawn from the results:

• Calculation of the heat field up front and distributing it in memory of a cluster computer is a feasible approach for using numerical heat data in combination with crystal growth calculations.
• Analyzing the solidification conditions resulting from the thermal field at different build heights suggests a planar solidification front. Nucleation under these conditions is occurring between diverging grains.
• In the horizontally oriented struts of the lattice structure without displacement of the shape to be melted, both predicted grain structures with and without nucleation matched the experimentally derived results, i.e., nucleation is not important here because enough well-oriented grains are available from the strut shell.
• The predicted grain structures for the vertically oriented struts differ between the cases with and without consideration of nucleation. The grain size measurements for the case with nucleation were significantly higher in accordance with the experiments. It is supposed that nuclei are forming in the gap between diverging grains.