# Construction of Cellular Substructure in Laser Powder Bed Fusion

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}.

## 1. Introduction

## 2. Materials and Methods

^{−1}, respectively. The hatch distance was set to be 800 μm, and the thickness of the metal powder layer was 50 μm. A cubic sample was fabricated with the point distance of 50 μm, the hatch distance of 60 μm, the layer thickness of 30 μm and the zigzag scanning strategy. The samples were mounted and polished (9 and 3 μm diamond suspension, and 0.05 μm colloidal silica suspension). The as-polished samples were finally polished by a 0.06 μm colloidal silica suspension using Vibromet 2 for 1 h. To show the structure of the substructure, the section normal to the scanning direction from the cubic sample was electrolytically etched at 6 V for 40 s using a 10% oxalic acid aqueous solution.

_{3}:H

_{2}O = 1:1:1. The morphology of the substructure was characterized by scanning electron microscope (SEM, TESCAN, Brno, Czech Republic), and the orientation map was collected by electron backscattered diffraction (EBSD, Oxford Instrument, Oxford, UK) at a step size of 0.5 μm.

## 3. Results and Discussion

#### 3.1. 3D Morphology of the Substructure

#### 3.2. Model of the Substructure

_{P}}, is built on the prism. Three Bunge Euler angles (φ

_{1}, Φ, φ

_{2}) can define a rotation from {K

_{P}} to a sample coordinate system (X’Y’Z’), {K

_{S}} [12]. As seen in Figure 2d, when intercepting the prism with a section of (X’OY’), a red hexagon appears in this section. The equation representing the section of (X’OY’) in {K

_{P}} is:

_{1}·sinΦ·x − cosφ

_{1}·sinΦ·y + cosΦ·z = 0.

_{1}= 1 + (p + q)

^{2}

_{2}= 1 + 4p

^{2}

_{3}= 1 + (p − q)

^{2}

_{1}·tanΦ

_{1}·tanΦ

_{1}, Φ).

_{1′}, L

_{2′}, L

_{3′}). By recombining Equations (2a)–(2e), the values of p and q are obtained by solving the following equations:

^{2}+ q

^{2}+ 2pq + 1 − m(1 + 4p

^{2}) = 0

^{2}+ q

^{2}− 2pq + 1 − n(1 + 4p

^{2}) = 0

_{1′}/L

_{2′})

^{2}

_{3′}/L

_{2′})

^{2}

_{1})/√3

_{1}is in the range of [0, $\frac{\pi}{3}$). Given a real hexagon, Equation (4) will return one solution to φ

_{1}.

_{1})

_{2}relies on the cardinal direction of the hexagon. On the one hand, the cardinal direction of a real hexagon is measured as the angle between the longest diagonal line and X’-axis of the section, i.e., in {K

_{S}}. The cosine value of the angle is labeled as CA. On the other hand, the direction of X’-axis of {K

_{S}} can be expressed as follows in {K

_{P}}:

**x**= (cosφ

_{1}·cosφ

_{2}− sinφ

_{1}·sinφ

_{2}·cosΦ, sinφ

_{1}·cosφ

_{2}+ cosφ

_{1}·sinφ

_{2}·cosΦ, sinφ

_{2}·sinΦ)

**l**= (−1, √3, (sinφ

_{1}+ √3cosφ

_{1})tanΦ)

**x**and

**l**:

**x**·

**l**/||

**l**|| = CA

_{2})

^{2}+ (cosφ

_{2})

^{2}− 1 = 0

_{2}.

_{2}, but not the unique solution. It is likely to determine the unique solution in consideration of the local thermal condition of the substructure. The growth direction of the prism in {K

_{S}} is the direction of Z-axis of {K

_{P}}, which is the third column of the rotation matrix in Equation (11):

**v**= (sinφ

_{2}·sinΦ, cosφ

_{2}·sinΦ, cosΦ)

**v**are checked to see whether they agree with the local thermal condition. The sign of the third component, cosΦ, can be determined from the scanning direction. Hence, the possible Φ in Equations (5a–d) are reduced from four to two. For each Φ, two possible φ

_{2}are obtained from Equation (8a) and (8b), and the possible vs. are calculated by Equation (9). The second component of

**v**, cosφ

_{2}•sinΦ, must be positive to ensure the upward growth of the cellular substructure. The sign of the third component of

**v**can be opposite to the sign of x’ due to heat conduction. It means that if the substructure locates on the left side, it shall grow towards the right side. Herein, the unique Φ, φ

_{2}, and

**v**can be selected from the possible mathematical solutions.

_{1}

^{C}, Φ

^{C}, φ

_{2}

^{C}), corresponding to the same area, are read from the EBSD map, as seen in Figure 2b. The crystallographic orientation of

**v**is:

**cv**= g·

**v**

_{S}}, to the crystal system, {K

_{C}}. The rotation matrix is calculated as follows [12]:

**v**have been determined, the crystallographic orientation corresponding to the growth direction can be easily obtained with the EBSD technique. The location information of the selected zone can be quickly read from the etched section, which can be further related to the local solidification parameters [16]. This model can also be more efficient than TEM in obtaining a large number of crystallographic orientations.

#### 3.3. Application of the Model

^{−1}tends to grow with the <001> crystallographic orientation. In the other two tracks, the crystallographic orientation of the substructure is rather random, and no obvious tendency to be around <110> or <111>. As seen in the subplot in the PFs, the morphologies of the substructure do not show any branching, and they are cell-like.

^{−1}, the substructure in the track tends to grow along with the <001> crystallographic orientation, which is more likely to be cell-like dendrite. At a higher scanning velocity, there can be no preferred crystallographic orientation, and the substructure is more likely to be that of a cell. On the one hand, the direction of the thermal gradient at different locations of the melting pool is different [6,7,22]. On the other hand, a polycrystalline substrate is used. Given the direction of the thermal gradient at a fixed location in the melting pool, the crystallographic orientation corresponding to the thermal gradient can vary as the melting pool moves [22]. Since the cell grows along the direction of the thermal gradient, the corresponding crystallographic orientation can be random.

^{−1}, there are also some points far away from the <001> crystallographic orientation, which may be cells. Besides, all the substructure in the other two tracks at the higher scanning velocity is cell. It is estimated that the critical growth rate of the transition may be around 0.31 ms

^{−1}in the single track with power of 200 W. This critical growth rate is also comparable in magnitude with the experimental and theoretical results [14,23]. It is noted that the critical growth rate in the present case is comparable to that obtained from the commonly used parameters in LPBF [2,4,5,6,7,8,11,27,28], and in other rapid solidification processes [14,15,16,23]. In these cases, it is suggested to be cautious to assume that the cellular substructure grows along the <001> crystallographic orientation. Besides, the development of the substructure may also depend on the as-solidified substructure in LPBF [7]. Further research may be achieved with the model presented in this research.

## 4. Conclusions

^{−1}and 1.25 ms

^{−1}, the growth direction of the substructure is random, and it tends to be along BD with the increase of scanning speed. The transition from cell-like dendrite to the cell is identified by the crystallographic orientation. At low scanning velocity, the substructure is cell-like dendrite growing along with the <001> crystallographic orientation. At high scanning velocity, the substructure is cell, whose crystallographic orientation is also random. The critical growth rate of the transition in the single track with power of 200 W can be around 0.31 ms

^{−1}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The substructure in the cubic sample. (

**a**) Low magnification. (

**b**–

**d**) High magnification. The boundaries of the colonies are marked out by the black dotted line. (

**b**) The substructure is inclined to the section. (

**c**) The substructure is parallel the section. (

**d**) The substructure is perpendicular to the section.

**Figure 2.**The process to calculate the growth direction of the substructure in the single track. (

**a**) is the etched transverse cross section and (

**b**) is the EBSD map corresponding to the section. The boundary of the track is marked out by the black dot line. (

**c**) is the region sampled from the transverse cross section, and the outline of the substructure is light gray. A real hexagon is selected from this region, whose boundary is marked out by black line. (

**d**) is the model of a hexagonal prism with the top surface of a regular hexagon. (

**e**) is the projected figure of the sampled region. (

**f**) is the transformative hexagon sectioned from the hexagonal prism according to the growth direction.

**Figure 3.**The stereographic pole figures of the growth direction and the stereographic inverse pole figures of the corresponding crystallographic orientation. Both (

**a**) and (

**d**) are calculated from the substructure in the track at scanning velocity of 0.31 ms

^{−1}. Both (

**b**) and (

**d**) are at the scanning velocity of 0.63 ms

^{−1}. Both of (

**c**) and (

**e**) are at the scanning velocity of 1.25 ms

^{−1}. In the pole figures, the inward growth directions are plotted as red points, while the outward growth directions are plotted as blue points.

Element | Fe | Cr | Ni | Mo | Mn | Si | N | O | P | C | S |

wt pct | balance | 16 to 18 | 10 to 14 | 2 to 3 | 2 | 1 | 0.1 | 0.1 | 0.045 | 0.03 | 0.03 |

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**MDPI and ACS Style**

Wang, Y.; Zhang, C.; Yu, C.; Xing, L.; Li, K.; Chen, J.; Ma, J.; Liu, W.; Shen, Z.
Construction of Cellular Substructure in Laser Powder Bed Fusion. *Metals* **2019**, *9*, 1231.
https://doi.org/10.3390/met9111231

**AMA Style**

Wang Y, Zhang C, Yu C, Xing L, Li K, Chen J, Ma J, Liu W, Shen Z.
Construction of Cellular Substructure in Laser Powder Bed Fusion. *Metals*. 2019; 9(11):1231.
https://doi.org/10.3390/met9111231

**Chicago/Turabian Style**

Wang, Yafei, Chenglu Zhang, Chenfan Yu, Leilei Xing, Kailun Li, Jinhan Chen, Jing Ma, Wei Liu, and Zhijian Shen.
2019. "Construction of Cellular Substructure in Laser Powder Bed Fusion" *Metals* 9, no. 11: 1231.
https://doi.org/10.3390/met9111231