Finite Difference Modelling of Laser Ablation Process: Iron Plate Simulation

LM Advanced Manufacturing
Finite Difference Modelling of Laser Ablation Process

1. Introduction
Pulsed Laser Ablation (PLA) process utilizes short laser pulses to deliver energy into the material and
remove a portion of it as vapour. Micro-, and nano- sized features can be obtained in most
materials, depending on process parameters, such as laser spot radius and pulse duration. The
highest quality of machined structures is achieved using femtosecond lasers, which yield minimal
thermally induced defects and the smallest heat affected zone (in metals) due to their extremely
short pulse. However, such lasers are associated with high acquisition and operating costs. [1] The
main advantages of PLA over traditional mechanical machining of microscopic features are higher
precision, robustness, and flexibility. [2]
PLA process exhibits highly localised (temporally and spatially) interaction of the material and the
laser beam. Consequently, heating and cooling rates are extremely high, often reaching ,
and the volume of the material subjected to thermal loads is small, making this process suitable for
processing brittle and thermally sensitive materials. [1] Processes based on the principles of PLA are
widely used in nano-patterning, cleaning of delicate surfaces, machining of microcomponents, and
various applications in biotechnology and medicine, such as manufacturing of implants or laser
surgery. [1, 3] It is also one of the methods of fabricating nanoparticles. [4]
Understanding of the underlying physical phenomena in PLA is crucial to be able to predict the
outcome of the laser-material interaction. Simulations based on mathematical models can be used
to aid the observation and analysis of different factors influencing PLA, and ultimately to optimise
the process.
The approach to modelling is based on a classical, transient heat conduction equation, with the
addition of a volume heat source representing the absorbed energy provided by the laser during a
pulse. Finite Difference approximation is used for the calculation. Laser light properties are assumed
to be those of a nanosecond Nd:YAG laser.
This project aims to model the temperature and material removal evolution in a thin iron plate,
caused by a single nanosecond laser pulse, and validate the results against numerical and
experimental results found in literature.
2. Methodology
2.1 Theoretical framework
The temperature change and distribution in the material during a laser pulse can be modelled by
solving numerically a two-dimensional heat conservation differential equation.
 (1)
where  
 is the density of iron,    is its specific heat capacity,  
the thermal conductivity. These are all assumed to be constant, because including the variations of
these properties with temperature observed in real materials would make the simulation too
complex for the purpose of this project. Variables in the equation are the temperature , time
, and the spatial dimensions representing the location along the length and depth of the plate
.
The last term in the equation (1), the volume heat source, represents the energy absorbed by the
material due to action of the laser. It is given in equation (2).
   (2)

 is the peak on-line laser intensity, which represents the peak power transmitted to the
area of the irradiated spot, given in equation (3).
 (3)
where  is the average power of the laser source,  the pulse frequency,  the pulse
duration and  the radius of the laser beam focus spot.
 is the attenuation coefficient, dependent on the given material’s extinction coefficient 
and the wavelength of the absorbed electromagnetic radiation , calculated using equation (4).
 (4)
The spatial and temporal profiles are assumed to be Gaussian and are represented by the term
.
 (5)
Values of all the material and laser properties used for the calculation are given in table 1.
Table 1: Material properties, [5] and laser parameters used in the model development.
Parameter
Value
2.2 Geometry, mesh, and chosen variable increments
A plate made of pure iron, of length    and thickness   , shown in Figure 1, is
modelled in two dimensions with the volume heat source located on the surface in the centre. The
selected spatial increments are    and   . This allowed discretisation of the
domain into  volumes, encircled by a mesh of 201 nodes in  and  directions, and a total
of  nodes. The time increment was found using equation (6).
 (6)
The obtained value of  is the largest possible time increment, which is required to prevent
numerical instability of the explicit solution of the finite difference approximation, which is described
in section 2.3. The time increment used in the simulation is    and was chosen to be two
Figure 1: Modelled plate, dimensions in mm. Light blue lines
represent the laser beam, whereas red and dark are X and Z axes.

orders of magnitude smaller than the calculated value for higher accuracy, balanced by a reasonable
computing time to solve the partial differential equation.
Nodes are indexed such as  signifies the temperature of the node  ( – step in the 
direction,  – step in the  direction) at the time step . The indices belong to one-dimensional
arrays:
2.3 Finite Difference Method (FDM)
FDM can be used to approximate the solution of partial differential equations. Explicit method is
used in this project. It requires less computational time than the implicit, while also being easier to
implement. In FDM, first order partial derivatives are replaced with the Forward Difference.
  (7)
Central Difference is used to approximate the second order derivatives.
 (8)
 (9)
Terms , , and  represent the truncation error of the approximation and are
neglected for the purpose of this simulation.
Equation (1) is substituted with appropriate terms obtained from equations (7), (8), (9), and
rearranged to give equation (10):
 (10)
For simplicity, constants can be group together to give a single parameter .
 (11)
Equation (10) is rearranged to give the final formula (12) used to calculate the temperature of the
node at the following time increment in the simulation.
  (12)
In the developed simulation, the volume heat source term is calculated separately for each node at
each time increment, using equation (13), and saved in a 3-dimensional array .
 (13)

Bibliography
[1]
M. Stafe, A. Marcu and N. N. Puscas, Pulsed Laser Ablation of Solids: Basics, Theory and Applications,
Berlin / Heidelberg: Springer, 2013.
[2]
J. Lopez, R. Kling, R. Torres, A. Lidolff, M. Delaigue, S. Ricaud, C. Hönninger and E. Mottay,
“Comparison of Picosecond and Femtosecond Laser Ablation for Surface Engraving of Metals and
Semiconductors,” Key Engineering Materials, vol. 496, no. 15, 2012.
[3]
M. N. R. Ashfold, F. Claeyssens, G. M. Fuge and S. J. Henley, “Pulsed laser ablation and deposition of
thin films,” Chemical Society Reviews, vol. 33, pp. 23-31, 2004.
[4]
M. Kim, S. Osone, T. Kim, H. Higashi and T. Seto, “Synthesis of Nanoparticles by Laser Ablation: A
Review,” KONA Powder and Particle Journal, 2016.
[5]
“MatWeb MATERIAL PROPERTY DATA,” Matweb, LLC, [Online]. Available:
http://www.matweb.com/search/DataSheet.aspx?MatGUID=654ca9c358264b5392d43315d8535b7d.
[Accessed 18th January 2021].