## Abstract

Powder spreading precedes creation of every new layer in powder bed additive manufacturing (AM). The powder spreading process can lead to powder layer defects such as porosity, poor surface roughness and particle segregation. Therefore, the creation of homogeneous layers is the first task for optimal part printing. Discrete element methods (DEM) powder spreading simulations are typically limited to a single layer and/or small number of particles. Therefore, results from such model configurations may not be generalized to multiple layer processes. In this study, a computationally efficient multi-layer powder spreading DEM simulation model is proposed. The model is calibrated experimentally using static Angle of Repose measurements. The adhesion model parameter, cohesive energy density is related to adhesive surface energy and strain energy release rate parameters. The model results show that interaction between particle and the powder spreading rake leads to noticeable variation in packing density, surface roughness, dynamic angle of repose (AOR), particle size distribution, and particle segregation. The powder model is experimentally validated using a recoater spreading rig to measure the dynamic AOR at spreading speeds consistent with recoating speeds and layer heights used in AM processes.

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## Acknowledgements

This research was supported by the High-Performance Computing for Manufacturing Project Program (HPC4Mfg), managed by the U.S. Department of Energy Advanced Manufacturing Office within the Energy Efficiency and Renewable Energy Office. This research used resources of the Compute and Data Environment for Science (CADES) at the Oak Ridge National Laboratory. The authors thank Dr. John Turner for support in preparation of this manuscript.

## Funding

This research is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

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## Appendix

### Appendix

Simplified Johnson–Kendall–Roberts (SJKR) adhesion model is derived by simplification of its original JKR form for contact of elastic spheres with adhesion [21, 30]. It is applicable for modeling weak interaction between surfaces in contact which in our case describes a multitude of possible interaction mechanisms that result in a weak cementation between the particles. A general form of the SJKR model describes the force, *F*_{c}, resisting separation between particles as:

where \( \kappa \) is an effective cohesive stress, termed cohesion energy in SJKR model, and *α* is an effective contact area between the particles. Different selection of \( \kappa \) and *α* result in variations of the model. In principle, parameter \( \kappa \) can be related to the cohesive surface energy between the particles, \( \Delta \gamma \), which is more commonly used in the continuum mechanics problems for modeling strong and weak interactions. The critical contact surface radius, *a*_{cr}, and pull out force, *F*_{cr}, in the original JKR model are:

where *R** denotes the effective particle radius of the contact of two particles, and *E** denotes the effective elastic contact modulus, as:

Symbols *R*_{i}, *E*_{i} and *ν*_{i} denote radius, elastic modulus and Poisson’s ratio of particle *i*, respectively. Selecting the effective contact surface *α* as the area of the critical contact surface for the pull-out force, *F*_{cr},

we have Eq. (10) after substitution of *a*_{cr} and *F*_{cr} from Eq. (5) and (6) into Eq. (9):

or equivalently

relate the two adhesive contact parameters *κ* and \( \Delta \gamma \).

Using Eq. (11), the nominal value of the cohesive energy density, *κ*, value of 3.6 × 10^{6} J/m^{3} (= 3.6 × 10^{7} erg/cm^{3}) used in our model is converted to the adhesive surface energy, \( \Delta \gamma \), value of 5.35 × 10^{−2} mJ/m^{2}. The calculated value for \( \Delta \gamma \) is fairly close to the value of 0.02 mJ/m^{2} for AOR = 29° and 0.06 mJ/m^{2} for AOR = 34° found in the literature for Ti–6Al–4V powder [16]. The value can be compared to the measured adhesive surface energy of 20–40 mJ/m^{2} in polymer materials [31].

An alternative theory for adhesion of particle was given by Maguis and Barquins [32]. Unloading of the spheres in contact was modeled as a crack propagation at the contact interface and the strain energy release rate for adhesion

Substituting *a*_{cr} and *F*_{cr} from Eqs. (5) and (6) into Eq. (12) for contact radius, *a*, and contact force, *F*, the cohesive energy density, *κ*, can be converted to the strain energy release. The cohesive energy density, *κ*, value of 3.6 × 10^{6} J/m^{3} is converted to *G* value of 5.93 × 10^{−3} mJ/m^{2} that is in the one order of magnitude lower as the adhesive surface energy converted value above, 5.35 × 10^{−2} mJ/m^{2}.

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Lee, Y., Gurnon, A.K., Bodner, D. *et al.* Effect of Particle Spreading Dynamics on Powder Bed Quality in Metal Additive Manufacturing.
*Integr Mater Manuf Innov* (2020). https://doi.org/10.1007/s40192-020-00193-1

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### Keywords

- Powder bed additive manufacturing
- Powder spreading
- Powder bed quality
- Multi-layer deposition
- Discrete element methods (DEM)
- Cohesive energy density