# Multi-Scale Modelling of the Bound Metal Deposition Manufacturing of Ti6Al4V

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Atomistic Modelling of Sintering Ti-6Al-4V Particles

#### Analysis of Diffusion

## 3. Microstructure Evolution

#### 3.1. Model Equations

^{2}, ${l}^{*}\sim $ $1.0$ $\mathsf{\mu}$$\mathrm{m}$, the value of $\delta $ is taken as ∼$1.0$ $\mathsf{\mu}$$\mathrm{m}$, which was shown in the previous part ${\delta}_{gb}$ ∼ $1.0$ $\mathsf{\mu}$$\mathrm{m}$ to be a natural length scale of the problem. The mobilities are defined as

#### 3.2. Mechanical Loading

#### 3.3. Results

#### Sintering Stress

## 4. Discrete Particle Model

## 5. Simple Model of Shrinkage

#### 5.1. Correlations

#### 5.2. Estimation of Shrinkage

#### 5.3. Fault Mitigation

## 6. Macroscopic Modelling of Sintering

#### 6.1. Equations for Modelling Metal-Fused Filament Fabrication Process

#### 6.2. Constitutive Relations for Olevsky Model

#### 6.3. Simulation Results for the Olevsky Model

#### 6.4. Constitutive Model for Backbone Polymer Debinding

#### 6.5. Nonlinear Shrinkage in Continuous Model

## 7. Summary and Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Image (obtained with OVITO [43]) of the initial state of sintering of 3 particles with HCP crystal structure ($\alpha $ phase). Diameters of the particles 1 to 3 (from left to right): 1st 14 nm, 2nd 12 nm, and 3rd 16 nm. The colours show Ti (red), Al (yellow), and V (blue). (

**b**) Distances between particles 1–2 (blue), 2–3 (red) and 1–3 (black) as a function of time during first 0.39 ns of sintering. Distance between pairs 1–2 and 2–3 is increased by 9 nm to be in scale with the distance for the pair 1–3.

**Figure 2.**(

**a**) Two interacting particles with the interface shown by the vertical dashed line. (

**b**) Distribution of titanium atoms in two particles along the x coordinate within a cylinder of radius 5 nm whose axis passes through the centres of both particles. The interface between the particles is shown by the vertical dashed line. (

**c**) Distance between particle centres as a function of time. (

**d**) The effective grain boundary width (blue squares) and neck radius (red circles) as a function of time. The ${\delta}_{GB}$ is shown by vertical dash-dotted red vertical lines.

**Figure 3.**A subset of sampled atomic trajectories. (

**left**) $xz$-view of trajectories of Ti atoms within the grain boundary layer after 1 ns of simulation. (

**right**) $yz$-view of the same trajectories. The color coding is the same as in Figure 2: red trajectories belong to the right particle while blue correspond to the left particle.

**Figure 4.**SEM micrographs of samples sintered at 1260 ${}^{\circ}$C with particle diameters of (

**a**) 0–20 $\mathsf{\mu}$$\mathrm{m}$, (

**b**) 20–45 $\mathsf{\mu}$$\mathrm{m}$, and (

**c**) 45–75 $\mathsf{\mu}$$\mathrm{m}$; sintered at 900 ${}^{\circ}$C for (

**d**) 0–20 $\mathsf{\mu}$$\mathrm{m}$, (

**e**) 20–45 $\mathsf{\mu}$$\mathrm{m}$, and (

**f**) 45–75 $\mathsf{\mu}$$\mathrm{m}$. Reprinted with permission from Springer Nature Customer Service Centre GmbH: Springer, International Journal of Minerals, Metallurgy, and Materials [6].

**Figure 5.**Results of the phase-field simulations of the microstructure evolution for 23 particles, in zero gravity, with periodic boundary conditions. (

**left**panel) Initial distribution of the 23 “particles”. (

**right**panel) Distribution of the metal particles in the end of the simulation time.

**Figure 6.**Results of the phase-field simulations of the microstructure evolution in 3D solid state sintering: (

**left**) at initial time and (

**right**) cross-sectional view after 24 time units.

**Figure 8.**(

**a**) Fragment of sliced NIST artefact for additive manufacturing. (

**b**) Zoomed view of a small part of the artefact. (

**c**) Geometry of the top few layers of the cylinder shown in figure (

**b**) obtained in KRATOS DEM. (

**d**) Discrete element model of the filaments in KRATOS DEM. The total number of spherical particles in the model ≈150,000.

**Figure 9.**The effect of gravity on the initial shrinkage obtained in the DEM model. (

**left**) initial state; (

**right**) final state after 10 s of simulation time. The results of simulations are well reproducible.

**Figure 10.**An example of the standard outcome of the simulations of the microstructure changes during debinding in zero gravity obtained using discrete element modelling of two rectangular cuboids: (

**left**) before shrinkage and (

**right**) after shrinkage.

**Figure 11.**(

**left**) Plot of the coordination number versus fractional density using tabulated data for minimum, maximum, random, compacted, sintered, and liquid phase sintered structures, including simulation data. The relation based on the square of the fractional density, according to Equation (15), is given by the solid line. Reprinted with permission from Elsevier and Copyright Clearance Center: Elsevier, Powder Technology [7]. Here LPS represents experimental data obtained using liquid phase sintering, while P+S corresponds to the experimental data obtained by pressing and sintering. “min and max” and “random” represent a mixture of experimental and simulated data. (

**right**) Experimental data for microhardness as a function of relative density [6]. The solid line represents the results of fitting by correlations (19).

**Figure 12.**Mechanical properties of the sintered samples as a function of the relative density. The (

**left**) panel shows experimentally obtained data for Young’s modulus [6,69] and the (

**right**) panel shows the experimental data for yield stress [6,68,69] as a function of the relative density in each case. The following symbols are used in the left panel ◯ represent results from [68] and ▽ corresponds to [6]; while in the right panel ▽—[6], ◯—[69], □—[68]. The solid lines represent the results of fitting by correlations (18).

**Figure 13.**Shrinkage as a function of initial ${f}_{in}$ and final ${f}_{fn}$ densities according to correlations given by Equation (17).

**Figure 14.**Dependence of the porosity on time at different values of the sintering stress. The initial porosity value is 0.3, $\eta =0.5\times {10}^{10}$ Pa$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s.

**Figure 15.**(

**a**) Porosity distribution at the end of sintering time; (

**b**) vertical displacement; (

**c**) horizontal displacement; (

**d**) stress distribution ${S}_{yy}$. (

**e**) ${S}_{xx}$; (

**f**) von Mises. The initial porosity value is ${\theta}_{0}$ = 0.3, $\eta =1\times {10}^{11}$ Pa$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s, ${P}_{L}=1$ MPa.

**Figure 16.**(

**a**) Concentration of the backbone debinder after sintering; (

**b**) total displacement; (

**c**) direction z displacement; (

**d**) x displacement. The parameters are $\beta ={10}^{-3}$ m${}^{3}$/kg, initial concentration ${10}^{3}$ mol/m${}^{3}$, diffusion coefficient $D=4\times {10}^{-6}$ m${}^{2}$/s, brown part density was taken as 3000 kg/m${}^{3}$, Young’s modulus 10 GPa, Poisson’s ratio 0.4.

**Figure 17.**COMSOL simulations of: (

**a**) the concentration of backbone debinder in the sintering part (after sintering); (

**b**) total displacement; (

**c**) directional strain; (

**d**) displacement in the vertical direction. The parameters were $\beta ={10}^{-3}$ m${}^{3}$/kg, initial concentration 10${}^{3}$ mol/m${}^{3}$, diffusion coefficient $D=4\times {10}^{-6}$ m${}^{2}$/s, brown part density was taken as 3000 kg/m${}^{3}$, Young’s modulus = 10 GPa, Poisson’s ratio = 0.4.

**Figure 18.**COMSOL simulations of: (

**a**) concentration of backbone debinder in the sintering part (after sintering); (

**b**) Von-Mises residual stress; (

**c**) directional displacement vertically; (

**d**) displacement in x. The parameters were $\beta ={10}^{-3}$ m${}^{3}$/kg, intial concentration ${10}^{3}$ mol/m${}^{3}$, diffusion coefficient $D=4\times {10}^{-6}$ m${}^{2}/s$, brown part density was taken as 3000 kg/m${}^{3}$, Young’s modulus = 10 GPa, Poisson’s ratio = 0.4.

# elt | lat | z | ielement | atwt | |||
---|---|---|---|---|---|---|---|

# alpha | b0 | b1 | b2 | b3 | alat | esub | asub |

# t0 | t1 | t2 | t3 | rozero | ibar | ||

Al | fcc | 12 | 13 | 26.9815 | |||

4.975 | 3.2 | 2.6 | 6 | 2.6 | 4.05 | 3.36 | 1.16 |

1 | 3.05 | 0.51 | 7.75 | 1 | 0 | ||

V | bcc | 8 | 23 | 50.942 | |||

4.89 | 4.74 | 1 | 2.5 | 1 | 3.04 | 5.3 | 1 |

1 | 1.7 | 2.8 | −1.6 | 1 | 0 | ||

Ti | hcp | 12 | 22 | 47.867 | |||

5.03 | 2.7 | 1 | 3 | 1 | 2.95 | 4.87 | 0.66 |

1 | 6.8 | −2 | −12 | 1 | 0 |

Parameter Name | Parameter Value |
---|---|

Density $\rho $ | 4500 kg/m${}^{3}$ |

Poisson ratio | 0.3 |

Coefficient of restitution | 0.2 |

Constitutive law | Hertz with JKR cohesion |

Young’s modulus, E | $5\times {10}^{6}$ Pa |

Cohesion ${E}_{c}$ | 1500 Pa |

Gravity acceleration | 9.8 m/s${}^{2}$ |

**Table 3.**Results of simulations of shrinkage of the submodel of the NIST artefact with the following parameters: $E={10}^{6}$ Pa; ${E}_{c}=500$ Pa; $\nu =0.3$; restitution = 0.2; $t=5$ s; porosity default; layout—uni (left) and cross (right).

E = 1 × 6 Pa; ${\mathit{E}}_{\mathit{c}}$ = 500 Pa; Default; Uni | E = 1 × 6 Pa; ${\mathit{E}}_{\mathit{c}}$ = 500 Pa; Default; Cross | ||||||
---|---|---|---|---|---|---|---|

LY0 | 17.3 | LZ0 | 12.34 | LY0 | 15.9 | LZ0 | 10.9 |

LYF | 15.29 | LZF | 9.31 | LYF | 13.96 | LZF | 9.01 |

LYF/LY0 | 0.884 | LZF/LZ0 | 0.754 | LYF/LY0 | 0.878 | LZF/LZ0 | 0.826 |

LX0 | 17.36 | LX0 | 26.42 | LX0 | 15.95 | LX0 | 23.53 |

LXF | 15.51 | LXF | 23.34 | LXF | 13.95 | LXF | 20.54 |

LXF/LX0 | 0.893 | LXF/LX0 | 0.883 | LXF/LX0 | 0.875 | LXF/LX0 | 0.879 |

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**MDPI and ACS Style**

Luchinsky, D.G.; Hafiychuck, V.; Wheeler, K.R.; Biswas, S.; Roberts, C.E.; Hanson, I.M.; Prater, T.J.; McClintock, P.V.E. Multi-Scale Modelling of the Bound Metal Deposition Manufacturing of Ti6Al4V. *Thermo* **2022**, *2*, 116-148.
https://doi.org/10.3390/thermo2030011

**AMA Style**

Luchinsky DG, Hafiychuck V, Wheeler KR, Biswas S, Roberts CE, Hanson IM, Prater TJ, McClintock PVE. Multi-Scale Modelling of the Bound Metal Deposition Manufacturing of Ti6Al4V. *Thermo*. 2022; 2(3):116-148.
https://doi.org/10.3390/thermo2030011

**Chicago/Turabian Style**

Luchinsky, Dmitry G., Vasyl Hafiychuck, Kevin R. Wheeler, Sudipta Biswas, Christopher E. Roberts, Ian M. Hanson, Tracie J. Prater, and Peter V. E. McClintock. 2022. "Multi-Scale Modelling of the Bound Metal Deposition Manufacturing of Ti6Al4V" *Thermo* 2, no. 3: 116-148.
https://doi.org/10.3390/thermo2030011