# Investigation of Ti65 Powder Spreading Behavior in Multi-Layer Laser Powder Bed Fusion
## 2 Methodology
### CFD Model  
To accurately replicate the powder spreading process in multi-layer PBF-LB, the influence of printed regions should be incorporated. The selective melting of the spread powder bed is simulated by a computational fluid dynamics (CFD) approach [12; 13]. The volume of fluid (VOF) method is used to track the evolution of the melt pool's free surface during laser melting. The model accounts for key physical phenomena influencing melt pool dynamics, including radiation heat transfer, the Marangoni effects, and vapor recoiling pressure resulting from metal evaporation. A Gaussian surface heat source is adopted to represent the laser heat source. The VOF method tracks the liquid-gas interface via the evolution of metal phase \(\alpha_{1}\), governed by [12; 13]  
\[\frac{\partial\alpha_{1}}{\partial t}+\
abla\cdot(\
u\alpha_{1})=0 \tag{11}\]  
\[\alpha_{1}+\alpha_{2}=1 \tag{12}\]  
where \(\alpha_{2}\) represents the volume fraction of the gas phase; \(t\) is the time; and \(\mathbf{v}\) is the velocity vector. The mass, momentum, and energy conservation equations are expressed as follows:  
(1) Mass:  
\[\
abla\cdot(\overline{p}\mathbf{v})=0 \tag{13}\]  
(2) Momentum [26; 27; 28]:  
\[ \begin{array}{l}\frac{\partial}{\partial t}(\overline{p}\mathbf{v})+\
abla\cdot( \overline{p}\mathbf{v}\otimes\mathbf{v})=\
abla\cdot(\overline{\mu}\
abla\mathbf{v})- \
abla p+\overline{p}\mathbf{\upvarepsilon}+F_{\mathrm{mushy}}\\ +(F_{\mathrm{tension}}+F_{\mathrm{Marangoni}}+F_{\mathrm{recoil}})|\
abla \alpha_{1}|\frac{2(\alpha_{1}\rho_{1}+\alpha_{2}\rho_{2})}{\rho_{1}+\rho_{2}} \end{array}  \tag{14}\]  
where \(\overline{\rho}\), \(\rho_{1}\), and \(\rho_{2}\) are the average density, metal density, and gas density, respectively; \(p\) is the pressure; and \(\overline{\mu}\) is the average viscosity. * The mushy zone damping force, \(F_{\mathrm{mushy}}\), models resistance to flow in partially solidified regions [26; 27]: \[F_{\mathrm{mushy}}=-\overline{\rho}K_{\mathrm{c}}\Bigg{[}\frac{(1-f_{\mathrm{ f}})^{2}}{f_{1}^{3}+C_{\mathrm{K}}}\Bigg{]}\mathbf{v}\] (15) \[f_{\mathrm{t}}= \begin{cases}1&T\geq T_{\mathrm{t}}\\ \frac{T-T_{\mathrm{s}}}{T_{\mathrm{t}}-T_{\mathrm{s}}}&T_{\mathrm{t}}\geq T\geq T _{\mathrm{s}}\\ 0&T\leq T_{\mathrm{s}}\end{cases} \] (16)  
where \(K_{\mathrm{c}}\) is the permeability coefficient; \(C_{\mathrm{K}}\) is a small constant to avoid division by zero;\(f_{\mathrm{t}}\) is the liquid fraction of the metal phase; \(T\) is the temperature; \(T_{\mathrm{s}}\) is the solidus temperature; and \(T_{\mathrm{t}}\) is the liquidus temperature. * Surface tension, \(\mathbf{F}_{\text{tension}}\), is modeled using the continuum surface force approach [29; 30]: \[\mathbf{F}_{\text{tension}}=\sigma\mathbf{n}=-\bigg{(}\sigma_{0}-\frac{\text{d}\sigma}{ \text{d}T}(T-T_{0})\bigg{)}\
abla\cdot\mathbf{n}\] (17) where \(\sigma\) is the surface tension coefficient; \(\frac{\text{d}\sigma}{\text{d}T}\) is the coefficient of surface tension varying with temperature; \(\sigma_{0}\) is the surface tension coefficient at the reference temperature \(T_{0}\); \(\kappa\) is the curvature of metal/gas interface; and \(\mathbf{n}\) is the unit vector. * The Marangoni force, \(\mathbf{F}_{\text{Marangoni}}\), accounts for surface tension gradients along the interface [26; 27]: \[\mathbf{F}_{\text{Marangoni}}=\frac{\text{d}\sigma}{\text{d}T}[\
abla T-\mathbf{n}( \mathbf{n}\cdot\
abla T)]\] (18)
* Recoil pressure, \(\mathbf{F}_{\text{recoil}}\), due to metal vaporization at high temperatures is given by [9; 31] \[\mathbf{F}_{\text{recoil}}=0.54P_{0}\exp\bigg{(}\frac{L_{\text{v}}m}{k_{\text{B} }}(\frac{T-T_{\text{v}}}{TT_{\text{v}}})\bigg{)}\mathbf{n}\] (19) where \(P_{0}\) is the ambient pressure; \(L_{\text{v}}\) is the latent heat of vaporization; \(m\) is the molecular mass; \(k_{\text{B}}\) is the Boltzmann constant; and \(T_{\text{v}}\) is the gasification temperature.