# Investigation of Ti65 Powder Spreading Behavior in Multi-Layer Laser Powder Bed Fusion
## 2 Methodology
### DEM Model  
The DEM is an effective approach to study the dynamic behavior and complex interaction of particulate systems; therefore, it is utilized to simulate the spreading process of Ti65 powder. The translational and rotational motions of each individual particle are governed by [PERSON]'s second law [21]:  
\[\left\{\begin{array}{l}{{m_{i}}\frac{{\rm d}{\mathbf{v}_{i}}}{{\rm d}t}=\sum{({\mathbf{ F}_{{\rm n},ij}}+{\mathbf{F}_{{\rm t},ij}})}+{m_{i}}{\bf g}}\\ {{l_{i}}\frac{{\rm d}{\mathbf{\omega}_{i}}}{{\rm d}t}=\sum_{j}{\mathbf{T}_{ij}}}\end{array}\right. \tag{1}\]  
where \(m_{i}\), \(\mathbf{\upsilon}_{i}\), \(\mathbf{\omega}_{i}\), and \(l_{i}\) represent the mass, translational velocity, angular velocity, and inertia moment of the particle \(i\); \(g\) is the acceleration of gravity; \({\mathbf{F}_{{\rm n},ij}}\) and \({\mathbf{F}_{{\rm t},ij}}\) are the normal and tangential forces between particle \(i\) and particle \(j\); and \({\mathbf{T}_{ij}}\) is the torque acting on particle \(i\).  
Notably, the weak adhesive inter-particle force becomes strong enough to have an impact on particle behavior when the particle size is less than 200 \(\upmu\)m [22] (the particle size of the Ti65 powder used in PBF-LB generally falls below 100 \(\upmu\)m). Therefore, the Hertz-Mindlin with Johnson-Kendall-Roberts (JKR) model [23] was employed to describe the contact forces between particles. The normal force is calculated as follows:  
\[{\mathbf{F}_{{\rm n},ij}}=-4(\pi E^{*}\gamma)^{\frac{1}{2}}a^{\frac{3}{2}}{\mathbf{n}} +\frac{4E^{*}}{3R^{*}}a^{2}{\mathbf{n}}-2\sqrt{\frac{5}{6}}\beta\sqrt{S_{{\rm n}} m^{*}}\mathbf{\upsilon}_{{\rm n}} \tag{2}\]  
where, in the first term (the adhesive force), \(E^{*}\) and \(\gamma\) are the equivalent Young's modulus and surface energy, respectively. The relationship between the radius of the contact area \(a\) and the normal overlap \(\delta_{{\rm n}}\) is given by  
\[\delta_{{\rm n}}=\frac{a^{2}}{R^{*}}-\left(\frac{4\pi\gamma a}{E^{*}}\right)^{ \frac{1}{2}} \tag{3}\]  
The second and third terms in Equation (2) denote the Hertz contact force and damping force in the normal direction. \(R^{*}\) is the equivalent radius. \(m^{*}\) and \(\mathbf{\upsilon}_{{\rm n}}\) are the equivalent mass and relative velocity between two contact particles in the normal direction. \(S_{{\rm n}}\) and \(\beta\) are the normal stiffness and damping coefficient, given by  
\[S_{{\rm n}}=2E^{*}\sqrt{R^{*}\delta_{{\rm n}}} \tag{4}\]  
\[\beta=-\frac{\ln e}{\sqrt{\ln^{2}e+\pi^{2}}} \tag{5}\]  
where \(e\) stands for the restitution coefficient. The tangential contact force is calculated by [24]  
\[{\mathbf{F}_{{\rm t},ij}}=-S_{{\rm t}}\delta_{{\rm t}}-2\sqrt{\frac{5}{6}}\beta \sqrt{S_{{\rm t}}m^{*}}\mathbf{\upsilon}_{{\rm t}} \tag{6}\]  
where \(\mathbf{\upsilon}_{{\rm t}}\) and \(\delta_{{\rm t}}\) are tangential relative velocity and displacement. \(S_{{\rm t}}\) represents the tangential stiffness, which is given by  
\[S_{{\rm t}}=8G^{*}\sqrt{R^{*}\delta_{{\rm n}}} \tag{7}\]  
The relationship between shear modulus \(G^{*}\) and [PERSON]'s modulus is given by  
\[G^{*}=\frac{E^{*}}{2(1+v)} \tag{8}\]The tangential force is limited by Coulomb friction force:  
\[F_{\mathrm{t}ij}=\begin{cases}F_{\mathrm{t}ij},&F_{\mathrm{t}ij}<\mu_{\mathrm{S}} F_{\mathrm{t},ij}\\ \frac{-\mathrm{p}}{\
u_{\mathrm{t}}}\mu_{\mathrm{S}}F_{\mathrm{n},ij},&F_{ \mathrm{t},ij}\geq\mu_{\mathrm{S}}F_{\mathrm{n},ij}\end{cases} \tag{9}\]  
The \(T_{ij}\) includes a torque resulting from tangential force and a torque from rotation [25]:  
\[T_{ij}=R_{ij}\times F_{\mathrm{t},ij}+\mu_{\mathrm{F}}F_{\mathrm{n},ij}R_{ij} \frac{\omega_{i}}{\omega_{i}} \tag{10}\]