# Numerical Simulation of Effect of Different Initial Morphologies on Melt Hydrodynamics in Laser Polishing of Ti6Al4V

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

_{cr}to predict the polishing effect of the polished surface within the spatial frequency domains [14,15,16]. When the spatial frequency amplitude of the polished surface is greater than the f

_{cr}, the surface roughness is significantly decreased. Vadali et al. extended the concept of one-dimensional critical frequency to the two-dimensional plane based on a series of physical equations that can predict the spatial frequency content and surface roughness after polishing [17]. The f

_{cr}was accurately obtained by solving the heat conduction differential equation by Ukar et al. [18]. It also predicted the polished surface morphology well. Further, Wang et al. developed a surface prediction model for thermocapillary regime smoothing [19]. They applied the capillary force prediction model proposed by Vadali et al. [17] to predict the spatial spectrum of polished surface by using introduced feature slope and normalized average displacement.

## 2. Numerical Simulation

#### 2.1. Governing Equations

- (a)
- The property of fluid phase fluid is treated as incompressible Newtonian laminar flow.
- (b)
- The material distribution satisfies continuity and isotropy.
- (c)
- The laser incident energy is considered as the surface heat flux.
- (d)
- Due to the ratio of the density of the liquid Ti6Al4V and the dynamic viscosity argon gas of is large, the influence of gas flow on the free surface evolution can be neglected.

_{V}is the body force of buoyancy and gravity of the molten pool [23].

_{ref}is the reference temperature, ρ

_{ref}is the reference density, β is the thermal expansion coefficient and g is the gravity constant.

_{p}is the specific heat, L

_{m}is the latent heat of melting, the definition of liquid fraction f

_{L}as follow [22,23,24,25,26]

_{s}is the solid phase temperature, T

_{l}is the liquid phase temperature.

#### 2.2. Model Geometry

#### 2.3. Boundary Conditions

- (1)
- Heat transfer boundary condition

_{a}is the ambient temperature, h is the convective coefficient.

- (2)
- Momentum boundary condition

_{r}and u

_{z}is the fluid flow velocity along the r and z directions.

- (3)
- Free surface boundary condition

#### 2.4. Laser Moving Heat Source

_{0}is the spot radius, r is the independent variable in the cylindrical coordinate system, v and t are the laser moving velocity and time, 0.05 (unit: mm) is the starting position of the laser polished surface. Furthermore, the product of piecewise function f(r*) and Q

_{s}describe the laser energy density acting within the laser beam. In addition, the modeling process parameters and specific boundary conditions set in physical fields are shown in Table 2 and Table 3, respectively.

#### 2.5. Moving Mesh

_{m}is the moving velocity of the mesh, u

_{mat}is the material velocity.

#### 2.6. Mesh and Configurations

## 3. Experimental Setup and Methods

#### 3.1. Polishing Experimental Setup

^{2}of less than 1.3 [37]. The dynamic focusing system (Model: SDL-F20PRO-3, from Suzhou FEELTEK Laser Technology Co., Ltd., Suzhou, China) with a maximum polishing area of 600 × 600 mm

^{2}can obtain a laser beam with a maximum scanning speed of 4000 mm/s, with the focal point of the laser beam generated at 720 mm from the polished surface [36]. Additionally, argon gas, with a purity of 99.99%, is used as a shielding gas to fill the processing chamber to prevent surface oxidation during the polishing process.

#### 3.2. Experimental Methods

## 4. Results and Discussion

#### 4.1. Molten Flow Behavior of Model 1

#### 4.2. Analysis of Temperature Field

#### 4.3. Analysis of Velocity Field

#### 4.4. Analysis of Free Surface Evolution

#### 4.5. Analysis of Secondary Surface Bumps Formed

#### 4.6. Molten Flow Behavior of Models 2 and 3

#### 4.7. Evolution of Melt Hydrodynamics for Models 1, 2 and 3

#### 4.8. Experimental Validation

## 5. Conclusions

- (1)
- The model demonstrated that the complex evolution of the melt hydrodynamics involving heat conduction, thermal convection, thermal radiation, melting and solidification during laser polishing.
- (2)
- The uniformity of the initial surface peaks and valleys distribution is positively correlated with the smoothing quality of the polished surface, but has less effect on temperature field, velocity field, as well as melt depth and width of the molten pool.
- (3)
- The surface rough profiles are not completely eliminated by capillary and thermocapillary forces due to the high cooling rate of the molten pool, resulting in the formation of secondary surface roughness. It was revealed that the short lifetime of the molten pool is the main reason why the surface bumps are not completely eliminated.
- (4)
- The numerical prediction of the depressions for Models 1, 2 and 3 are approximate 26 μm, 12 μm and 13 μm at about 1 mm on the polished surface. Accordingly, the experimental molten pool depths are about 24 μm, 14 μm and 15 μm as well as the errors are approximately 8.3%, 14.3% and 13.3%, respectively.
- (5)
- The model not only predicts the morphological evolution of different surfaces from rough to smooth in laser polishing, but also can be suitable for guiding the optimization of polishing parameters such as laser power and scanning speed. Additionally, this model can be applied to most metallic materials in laser polishing.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

LP | Laser polishing |

PAM | Plasma assisted milling |

FEM | Finite element method |

ALE | Arbitrary Lagrangian-Eulerian |

SSM | Surface shallow melting |

2D | Two dimensional |

f_{cr} | Critical frequency (Hz) |

ρ | Density (kg m^{−3}) |

t | Laser radiation duration (ms) |

T | Temperature (K) |

$\overrightarrow{u}$ | Velocity (m s^{−1}) |

K | Thermal conductivity (Wm^{−1} K^{−1}) |

${\overrightarrow{u}}_{m}$ | Mesh velocity |

u_{mat} | Material velocity (m s^{−1}) |

p | Pressure (Pa) |

I | Identity matrix |

Μ | Dynamic viscosity (Pa s) |

F_{V} | Body force (N m^{−3}) |

T_{ref} | Reference temperature (K) |

ρ_{ref} | Reference density (kg m^{−3}) |

β | Thermal expansion coefficient (K^{−1}) |

g | Gravity constant (N kg^{−1}) |

C_{p} | Specific heat (Jkg^{−1} K^{−1}) |

f_{L} | Liquid fraction |

T_{s} | Solidus temperature (K) |

T_{l} | Liquidus temperature (K) |

T_{m} | Melting temperature (K) |

T_{b} | Boiling temperature (K) |

T_{a} | Ambient temperature (K) |

ρ_{s} | Solidus density (kg m^{−3}) |

ρ_{l} | Liquidus density (kg m^{−3}) |

k_{s} | Solidus thermal conductivity (Wm^{−1} K^{−1}) |

k_{l} | Liquidus thermal conductivity (Wm^{−1} K^{−1}) |

C_{p-s} | Solidus specific heat (Jkg^{−1} K^{−1}) |

C_{p-l} | Liquidus specific heat (Jkg^{−1} K^{−1}) |

H | Convective coefficient (Wm^{−2} K^{−1}) |

∂γ/∂T | Temperature derivative of surface tension (Nm^{−1} K^{−1}) |

L_{m} | Latent heat of melting (Jkg^{−1}) |

Ε | Emissivity |

α_{0} | Absorptivity |

σ | Stefan-Boltzmann constant |

u_{r} | Fluid flow velocity along the r direction |

u_{z} | Fluid flow velocity along the z direction |

γ | Surface tension coefficient (N m^{−1}) |

κ | Surface curvature (m^{−1}) |

$\overrightarrow{n}$ | Normal vector |

$\overrightarrow{t}$ | Tangential vector |

P | Laser power (W) |

r_{0} | Laser beam radius (m) |

M^{2} | Laser beam quality |

R | The r-component in the cylindrical coordinate system |

V | Laser moving velocity (mm s^{−1}) |

f(r*) | Laser radiation area |

Q_{s} | Stationary laser energy density (J cm^{−2}) |

Q_{m} | Moving laser energy density (J cm^{−2}) |

t_{h} | Laser heating duration (ms) |

t_{c} | Cooling duration (ms) |

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**Figure 1.**Geometry model of molten pool. (

**a**) Optical morphology of initial surface; (

**b**) Optical morphology of initial surface after Fourier filtering; (

**c**) Surface profile height of Model 1; (

**d**) Geometry of Model 1; (

**e**) Surface profile height of Model 2; (

**f**) Geometry of Model 2; (

**g**) Surface profile height of Model 3; (

**h**) Geometry of Model 3.

**Figure 3.**Experimental setup and device of the polishing on Ti6Al4V. (

**a**) Experiment device; (

**b**) Principle of experiment device.

**Figure 4.**Evolution of molten pool morphology and the distribution of temperature field (color surface contour, unit: K) and velocity field (colored arrow plots, unit: m/s) of Model 1.

**Figure 5.**The dominant of capillary and thermocapillary forces as well as viscosity distribution at 0.3 ms heating duration.

**Figure 6.**Evolution of local molten pool morphology and the distribution of temperature field (color surface contour, unit: K) and velocity field (colored arrow plots, unit: m/s) of Model 1.

**Figure 7.**Evolution of molten pool morphology and the distribution of temperature field (color surface contour, unit: K) and velocity field (colored arrow plots, unit: m/s) of Model 2.

**Figure 8.**Evolution of molten pool morphology and the distribution of temperature field (color surface contour, unit: K) and velocity field (colored arrow plots, unit: m/s) of Model 3.

**Figure 9.**Variation of the maximum velocity of liquid metal in the polished area with radiation duration.

**Figure 10.**Variation of molten pool depth and width in the polished area with radiation duration: (

**a**) melting depth; (

**b**) melting width.

**Figure 12.**The surface morphology of the single-line polished tracks and comparison between simulated surface height profile and experimental laser polished profiles for Models 1, 2 and 3. (

**a**) Polished optical morphology of Model 1; (

**b**) Surface profile height of Model 1; (

**c**) Polished optical morphology of Model 2; (

**d**) Surface profile height of Model 2; (

**e**) Polished optical morphology of Model 3; (

**f**) Surface profile height of Model 3.

Parameter | Nomenclature | Value |
---|---|---|

Solidus temperature (K) | T_{s} | 1877 |

Liquidus temperature (K) | T_{l} | 1923 |

Melting temperature (K) | T_{m} | 1900 |

Boiling temperature (K) | T_{b} | 3315 |

Ambient temperature (K) | T_{a} | 298.15 |

Solidus density (kg m^{−3}) | ρ_{s} | 4420 |

Liquidus density (kg m^{−3}) | ρ_{l} | 4000 |

Dynamic viscosity (Pa s) | μ | 0.005 |

Solidus thermal conductivity (Wm^{−1} K^{−1}) | k_{s} | 21 |

Liquidus thermal conductivity (Wm^{−1} K^{−1}) | k_{l} | 29 |

Solidus specific heat (Jkg^{−1} K^{−1}) | C_{p-s} | 670 |

Liquidus specific heat (Jkg^{−1} K^{−1}) | C_{p-l} | 831 |

Convective coefficient (Wm^{−2} K^{−1}) | h | 10 |

Temperature derivative of surface tension (Nm^{−1} K^{−1}) | ∂γ/∂T | −2.8 × 10^{−4} |

Latent heat of melting (Jkg^{−1}) | L_{m} | 2.86 × 10^{5} |

Emissivity | ε | 0.6 |

Absorptivity | α_{0} | 0.3 |

Polishing Parameter (Unit) | Nomenclature | Value |
---|---|---|

Laser beam radius (mm) | R_{0} | 0.135 |

Laser power (W) | P | 150 |

Laser scanning speed (mm s^{−1}) | v | 300 |

Laser heating duration (ms) | t_{h} | 3.5 |

Cooling duration (ms) | t_{c} | 0.3 |

Boundary Condition | Boundary (See Figure 1) | Physical Condition |
---|---|---|

Boundary heat source | 1 | Laser radiation |

Convection | 1, 2, 3 | Natural convection |

Diffuse surface | 1, 2, 3 | Radiation |

Thermal insulation | 4 | Insulation |

Capillary force | 1 | Weak contribution |

Themocapillary force | 1 | Marangoni effect |

Wall | 2, 3,4 | No slip wall |

Parameter (Unit) | Top Layer | The Rest |
---|---|---|

Maximum element size (μm) | 0.8 | 20 |

Minimum element size (μm) | 0.002 | 0.024 |

Maximum element growth rate | 1.05 | 1.1 |

Curvature factor | 0.2 | 0.2 |

Ti | Al | V | C | Fe | O | N |
---|---|---|---|---|---|---|

Balance | 5.50–6.75 | 3.50–4.50 | 0.08 | 0.30 | 0.20 | 0.05 |

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**MDPI and ACS Style**

Li, K.; Zhao, Z.; Zhou, H.; Zhou, H.; Yin, J.; Zhang, W.; Zhou, G.
Numerical Simulation of Effect of Different Initial Morphologies on Melt Hydrodynamics in Laser Polishing of Ti6Al4V. *Micromachines* **2021**, *12*, 581.
https://doi.org/10.3390/mi12050581

**AMA Style**

Li K, Zhao Z, Zhou H, Zhou H, Yin J, Zhang W, Zhou G.
Numerical Simulation of Effect of Different Initial Morphologies on Melt Hydrodynamics in Laser Polishing of Ti6Al4V. *Micromachines*. 2021; 12(5):581.
https://doi.org/10.3390/mi12050581

**Chicago/Turabian Style**

Li, Kai, Zhenyu Zhao, Houming Zhou, Hao Zhou, Jie Yin, Wei Zhang, and Guiyao Zhou.
2021. "Numerical Simulation of Effect of Different Initial Morphologies on Melt Hydrodynamics in Laser Polishing of Ti6Al4V" *Micromachines* 12, no. 5: 581.
https://doi.org/10.3390/mi12050581