Laser Ablation Modelling Using FEA

Laser ablation is a process in which a high-intensity laser beam removes material from a solid surface through various thermophysical mechanisms, including melting, vaporization, and plasma formation. Understanding and modeling laser ablation using Finite Element Analysis (FEA) requires solving complex heat transfer and phase transition equations to capture the dynamic response of the material under intense laser irradiation.

1. Governing Equations for Laser-Material Interaction

The laser ablation process involves energy absorption, heat conduction, phase change, and material ejection. The fundamental governing equation is the heat conduction equation with a source term representing the laser heat input:

$$\rho c_p\frac{\partial T}{\partial t}=\mathrm{\nabla }\cdot (k\mathrm{\nabla }T)+Q_{\text{laser}}$$

where:

• $\rho$ is the material density,
• $c_p$ is the specific heat capacity,
• $T$ is the temperature field,
• $k$ is the thermal conductivity,
• $Q_{\text{laser}}$ represents the absorbed laser energy per unit volume.

1.1 Laser Heat Source Term

The laser heat input can be modeled using a volumetric heat source based on the Beer-Lambert absorption law:

$$Q_{\text{laser}}(z,t)=\eta I_0{e}^{-\alpha z}$$

where:

• $\eta$ is the absorption coefficient,
• $I_0$ is the incident laser intensity,
• $\alpha$ is the optical penetration depth,
• $z$ is the depth from the surface.

For a pulsed laser, the intensity varies with time, and a Gaussian spatial profile is often assumed:

$$I(r,t)=I_0{e}^{-2\left(\frac{r^2}{w_0^2}\right)}f(t)$$

where:

• $w_0$ is the beam waist,
• $f(t)$ is the temporal pulse shape.

2. Phase Change and Material Removal

Material removal during laser ablation occurs through phase transitions such as melting and vaporization. These transitions require an accurate phase change model.

2.1 Melting and Solid-Liquid Interface Tracking

The Stefan condition governs the movement of the melting front:

$$\rho L_m{v}_n=k_s\frac{\partial T}{\partial n}-k_l\frac{\partial T}{\partial n}$$

where:

• $L_m$ is the latent heat of melting,
• $v_n$ is the normal velocity of the solid-liquid interface,
• $k_s$ and $k_l$ are the thermal conductivities of the solid and liquid phases.

2.2 Vaporization and Surface Recession

The vaporization process is described using the Clausius-Clapeyron relation:

$$P_v=P_0{e}^{-\frac{L_v}{R{T}_v}}$$

where:

• $P_v$ is the vapor pressure at the surface,
• $P_0$ is the reference vapor pressure,
• $L_v$ is the latent heat of vaporization,
• $R$ is the universal gas constant,
• $T_v$ is the vaporization temperature.

The material removal rate is then determined by the Hertz-Knudsen equation:

$$\dot{m}={\alpha }_v{P}_v\sqrt{\frac{M}{2\pi R{T}_v}}$$

where:

• $\dot{m}$ is the mass removal rate per unit area,
• ${\alpha }_v$ is the evaporation coefficient,
• $M$ is the molecular weight of the material.

3. Thermal Stresses and Shock Waves

High thermal gradients during laser ablation induce thermal stresses, which can lead to cracking or fragmentation. The thermal stress tensor is given by:

$${\sigma }_{ij}=E{\alpha }_T(T-T_0){\delta }_{ij}$$

where:

• $E$ is Young’s modulus,
• ${\alpha }_T$ is the thermal expansion coefficient,
• $T_0$ is the reference temperature,
• ${\delta }_{ij}$ is the Kronecker delta.

The ablation process can also generate shock waves in the surrounding gas due to rapid material ejection. The shock wave propagation is modeled using the Euler equations:

$$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{v})=0$$

$$\frac{\partial (\rho \mathbf{v})}{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{v}\mathbf{v})+\mathrm{\nabla }P=0$$

$$\frac{\partial E}{\partial t}+\mathrm{\nabla }\cdot ((E+P)\mathbf{v})=0$$

where:

• $\mathbf{v}$ is the velocity field,
• $P$ is the pressure,
• $E$ is the total energy per unit volume.

4. Surface Morphology and Recoil Pressure Effects

The vaporization process generates recoil pressure on the surface, influencing material ejection. The recoil pressure is given by:

$$P_r=\frac{P_v}{1+\frac{\gamma }{\gamma -1}{M}_v^2}$$

where:

• $\gamma$ is the heat capacity ratio,
• $M_v$ is the Mach number of the vaporized material.

This pressure can modify the surface morphology, leading to crater formation and splashing effects. The surface evolution can be tracked using a level-set method or volume-of-fluid approach.

5. Plasma Formation and Absorption

For high-intensity lasers, plasma formation occurs above the material surface, affecting energy absorption and shielding the substrate. The plasma absorption coefficient follows the Drude model:

$${\alpha }_p=\frac{e^2{n}_e}{{ϵ}_0{m}_e{\omega }^2}$$

where:

• $e$ is the electron charge,
• $n_e$ is the electron density,
• ${ϵ}_0$ is the permittivity of free space,
• $m_e$ is the electron mass,
• $\omega$ is the laser frequency.

The plasma temperature evolution is governed by the energy balance equation:

$$\rho c_p\frac{d{T}_p}{dt}=Q_{\text{abs}}-Q_{\text{loss}}$$

where:

• $Q_{\text{abs}}$ is the laser energy absorbed by the plasma,
• $Q_{\text{loss}}$ includes radiation and convection losses.

Applications of Laser Ablation Modelling

• Material Removal and Damage Analysis: FEA helps in understanding the effects of laser ablation on materials, including thermal damage, buckling failure, and changes in mechanical properties2.
• Medical Applications: Models can be calibrated against experimental data to predict crater geometry and volume in applications like lithotripsy1.
• Industrial Processes: Simulations aid in optimizing laser parameters for efficient material processing and minimizing thermal stress3.

Challenges and Future Directions

• Complexity of Physical Phenomena: Current models often simplify complex phenomena like laser scattering and melt pool formation. Future work may focus on incorporating these effects for more accurate predictions1.
• Material Properties: Accurate simulation requires detailed knowledge of material properties at high temperatures, which can be challenging to obtain2.

Modeling laser ablation using FEA requires solving a combination of heat transfer, phase transition, fluid dynamics, and plasma physics equations. Accurate boundary conditions, such as heat flux, recoil pressure, and vaporization mass loss, are essential for capturing the realistic behavior of the process. Incorporating multi-physics effects, such as thermal stresses and plasma shielding, enhances the predictive capability of the model for applications in microfabrication, medical treatments, and industrial machining.

I hope this guide will definitely help you to get. overall idea how things work If you are working with 1D or analytic calculation, you can easily use python to create your model for 2D or 3D you may need some specialized FEA software tools. If you want to discuss, feel free to talk HERE.

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