2D thermo-physical modelling and simulation on LIPAA of moving heat source for microchannel fabrication on polycarbonate
5.3 Thermo-physical modelling of LIPAA using Finite Element Method (FEM)
In the study, a two-dimensional nonlinear transient finite element model of Laser-Induced Plasma Assisted Ablation (LIPAA) of PC has been developed. A temperature profile is generated by solving the finite element model considering the temperature dependent material properties of the PC. The influence of the plasma and the input laser irradiance have been taken into consideration in the heat flux model of the simulation. The developed model is employed to study the effects of laser parameters on the channel dimensions during the LIPAA process. The model takes into account only the melting of the material.
As mentioned in the previous chapter, ablation on both the PC sheet and the aluminium target is obtained in the LIPAA process. However, our present study is confined to the PC sheet only. The model is applied not only to analyze the effect of the parameters on LIPAA of PC, but also on the effect of the moving heat flux on the PC sheet.
5.3.1 Assumptions
In the present work, certain assumptions were made to make the model realistic. These are as follows:
The heat source is assumed to be of Gaussian shape (Acherjee et al., 2012).
Considering Gaussian-shaped heat flux at the zone of irradiation will make the model conditions closer to the real LIPAA phenomenon.
The thermal properties of the material are temperature dependent.
The reflectivity of both the PC and aluminium is considered.
Only the melting of the PC above its melting poit temperature is considered. The decomposition of the PC above its degradation temperature is neglected.
The transparent material and metal target are isotropic and homogeneous in nature.
The ambient temperature is 300 K.
5.3.2 Geometric model
In the LIPAA process, as shown in figure 5.2(a), the PC sheet is arranged above the aluminium sheet without the application of any external pressure. Gaussian heat flux is considered to be moving along a straight line at the interface of the two materials. A section ABCD on the x‒z plane is considered at the zone of irradiation of the moving laser, representing the two-dimensional domain of the LIPAA process. However, for the numerical study, only an axisymmetric section of the PC from the two-dimensional domain is considered as represented in the figure. 5.2(b). It is because a 2D domain of the model consists of much fewer elements, which reduces the simulation time. Further, the temperature profile generated in the axial and longitudinal direction is sufficient to predict the channel width and channel depth in much less time.
Figure 5.2 Geometric representation of the LIPAA model
In the present numerical model, geometric model of PC has been assigned with the essential material properties. The thermal properties of the PC are considered temperature dependent, as shown in table 5.1. Other thermal and physical properties of the PC and the aluminium were considered constant and are listed table 5.2.
Table 5.1Temperature dependent properties of PC
Temperature (K) Conductivity (W/mK) Specific Heat Capacity (J/kg K)
323 0.204 1330
373 0.213 1370
423 0.213 1480
473 0.215 1500
523 0.226 1540
573 0.238 1550
Table 5.2Thermal and physical properties of PC and aluminium (Acherjee et al., 2012;
Sharma et al., 2019)
Property Value
Polycarbonate
Density (kg/m3) 1200
Emissivity 0.95
Reflectivity 0.07
Melting Point (K) 503
Aluminium Reflectivity 0.96
Density (kg/m3) 2700
5.3.3 Thermal analysis
The two-dimensional non-linear transient heat conduction equation that governs the heat generation during the LIPAA process can be described as:
T T T
k k c
x x z z t
(5.1)
where x, y are the Cartesian coordinates, t is the time, k is the thermal conductivity of the material, ρ is the density, c is the specific heat of the material and T is the temperature.
The boundary conditions include the moving Gaussian heat flux, convection and radiation on the thermally affected boundary of the PC. The other two sides of the axisymmetric domain are taken to be insulated. The boundary conditions on the axisymmetric section of the PC have been represented in figure 5.3 (concerning figure 5.2(a)).
Figure 5.3 Boundary conditions for the numerical model Mathematically, the boundary conditions can be expressed as:
k T q
x
for pulse-on time (5.2)
=0 for pulse-off time and
4 4
s s
( ) ( )
T T
k k h T T T T
x z
(5.3)
where q is the laser heat source, h is the convective heat transfer coefficient, Ts is the surface temperature, ε is the emissivity of the transparent material and σ is the Stefan Boltzmann constant (5.67 × 10-8 W/m2K4). By merging the convection and the radiation conditions, the final boundary condition can be expressed as (Acherjee et al., 2012)
r( s )
T T
k k h T T
x z
(5.4)
where hr is the combined heat transfer coefficient. The combined heat transfer coefficient is given by
2 2
r ( s ) ( s )
h h
T T T T (5.5)Initially, i.e. at t = 0, the workpiece is considered to be at ambient temperature, T∞
of 300 K i.e.
( , , 0)
T x y T (5.6)
5.3.4 Laser heat source
The heat source is considered taking into account the plasma power density besides the input laser intensity. Assuming that the laser intensity follows Gaussian distribution, the heat source, q can be expressed as
a 1
(1 )
q R I q (5.7) where Ra is the reflectivity of the metal target, I is the laser pulse intensity and q1 is the plasma power density.
The Gaussian distribution of the laser pulse intensity is given by Sundqvist et al.
(2017) as:
2 oexp 2 x2
I I
R
(5.8) where Io is the peak laser pulse intensity and R is the laser spot radius. The plasma power density can be estimated by using the following relation:
1
plasma on
q E
t (5.9)
where Eplasma is the plasma energy density and ton is the pulse duration.