S3.1 Discussion on the complete modeling of LMP

In level 1 and level 2 of chapter 3 we have covered many aspects on the modelling of Laser Machining Processes. It's time for a general review, then we will move onto a new level.

The first step in modelling is the discription of laser energy. We discussed this topic in Section 3.2 Level 1. In practice, we need to measure the pulse energy or average power of the laser beam, then measure the beam spot size along the optical axis. Usually the beam quality M² is of great interest. then we can calculate the other beam indexes such as depth of focus, beam waist. We get the most general form of laser energy model:

Where I₀(t, z) is function of time and distance from beam waist z, SP(x,y) is the spatial distribution of laser energy and it is a function of x and y. SP may take forms other than Gaussian distribution, we can even integrate complex spatial modulation into this function, for example, pattern information in texturing can be embeded in this function.

The second step is the characterization of Laser energy coupling with the target material. The physical processes in laser material interaction is important for understanding the capabilities and limitations of laser machining processes. When laser beam strikes on the target material, part of the energy is reflected and part of it is absorbed, the absorbed energy heats up the target materials and phase change of the target material is induced. This has been discussed in Section 3.3 Level 1. For a Gaussian beam with beam radius r and for a mterial with surface absorptivity A=1-R (R is the surface reflectivity), the laser intensity inside the material is:

I(x,y,z,t)=(1-R)I0(t)exp(-a*z)exp(-(x^2+y^2)/r^2)

Where z is the distance from the top surface, a is the damping coefficient with depth according to Lambert's law.

The reflection coefficient R for normal angles of incidence from air to opaque perfect flat clean metal surface can be computed using the following formula:

R=[(1-n)²+k²]/[(1+n)²+k²]

and the absorptivity A of opaque metal surface is:

A= 1 – R =4n/[(n+1)²+k²]

Where n is the refraction coefficient of material, k is the extinction coefficient of material. Both value can be looked up in handbooks. We list some of the values in the following table.Remember these optical properties are functions of radiation wavelength and varies with temperature.

Reflectivity varies with both the angle of incidence and plane of polarization. If the plane of polarization is in the plane of incidence, the ray is called parallel ray ("p" ray); if the plane of polarization is perpendicular to the plane of incidence, the ray is called "s" ray. The reflectivity coefficients for perfect flat surfaces of the "p" ray and "s" ray are:

R_p=[(n-1/cosf )²+k²] / [(n+1/cosf )²+k²]

R_s=[(n-cosf )²+k²] / [(n+cosf )²+k²]

The readers are refered to Section 2.9 Level 2 for a detailed discussion laser beam reflection and absorption.

The third step is looking up the necessary properties of the materials. Since laser machining process covers a large temperaure range, the variation of material property with temperature should be considered. For example, the heat conductivity and heat capacity are used in the energy equation of the model and their change with temperature should be considered. When temperature or other effects, such as pressure, are considered, the governing equation becomes high nonlinear, as a result, iterations have to be used for computation.

The commonly used material properties are:

Energy balance analysis: Density, heat capacity, specific heat ratio, heat conductivity, heat diffusivity, latent heat, melting point, vaporization point, etc.;

Stress and momentum analysis: viscosity, Young's modulus, shear modulus, Poison ratio, stress-strain constitution relation, etc.

Example, properties of copper: Density r = 8960 kg/m3, melting temperature Tm = 1083°C, latent heat of melting Lm = 13.0 kJ/mol at 1atm, vaporization temperature Tv = 2543°C, and latent heat of vaporization Lv = 302 kJ/mol at 1atm. Other thermal properties of copper should be treated as temperature sensitive and be interpolated from tabulated data in order to obtain reasonable calculation results. The isotropic specific heat capacity is given by

The thermal conductivity for solid copper is

and for liquid copper

One comment: the process of laser machining is usually very fast, strain rate effects, work hardening, visco-elasticity-plasiticity, temperature dependence in the stress strain constitution relation should be considered for accurate modelling. When high pressure is involved, such as in laser shock processing, the influence of pressure on the material stress-strain relation should also be considered.

The fourth step is to analyze and simplify the physical phenomena, extract the Governing equations.

The real physical processes in laser machining are very complex. Many research work have been done based on various simplifications.

Many models of laser drilling have been developed. Paek developed a theoretical model to predict the temperature profile assuming a laser beam of circular cross section and uniform intensity (Paek and Gagliano, 1972). Dabby calculated the transient temperature and penetrating velocity during the vaporization process (Dabby and Paek, 1972). The models more recently developed (Ho, et al., 1995; Kar and Mazumder, 1994) considered effects of gas dynamics and Knudsen layer discontinuity during the ablation process. These models assume 1D heat transfer in target material, recognizing that the machining depth is much smaller than the diameter of hole, which is reasonable for relatively large holes (a few hundred microns). As a result, however, the effects of beam profiles and cavity profiles are not considered. These factors are important when the size of the hole is comparable to the drilling depth. Modest developed a transient three-dimensional heat conduction model for material volume being machined (Modest, 1996). However, the model assumes that vaporization occurs in a single step without melting. Gas dynamics and discontinuity layer were not taken into account. This is not suitable for laser machining of metals on nanosecond time scale. Other models have been developed to study the phenomena of vapor plume and plasma during laser-plasma-solid interactions (Aden, et al., 1992; Singh and Narayan, 1990), some of which were developed with the application of thin-film deposition via laser ablation in mind. Many modelling work has also been done on laser cutting process.

In general, modelling efforts can be divided according to their major interests, some are foused on the target material, some are on gas dynamics and some are on the plasma phenomena. We have covered some of the simple models in level one and two. The governing equation are usually Energy equation and momentum equations, if mass transfer is important the mass transportation equation may also be part of it. In level three we will introduce you to more general models.

The fifth step is to setup Boundary conditions and start programming. Boundary conditions are important part of the model, they influence the programming and calculation results greatly. One can write programs to solve their special problems, or resort to commercial packages. These will be illustrated in detail in the following sections.

References:

Aden, M., et al., 1992, "Laser-induced vaporization of a metal surface," J. Phys., D 25, pp. 57-65.

Dabby, F.W., and Paek, U.C., 1972, "High-intensity laser-induced vaporization and explosion of solid material," IEEE J. of Quantum Electronics, Vol. QE-8, No. 2, pp. 106-111.

Ho, J. R., et al., 1995, "Computational Model for the Heat Transfer and Gas Dynamics in the Pulsed Laser Evaporation of Metals," J. Appl. Phys., Vol. 78(7), pp. 4696-4709.

Kar, A., and Mazumder, 1994, "Mathematical model for laser ablation to generate nanoscale and submicometer-size particles," J. Physical Review E, Vol. 49(1), pp. 410-419.

Kezhun Li and Paul Sheng, 1995, "Computational model for laser cutting of steel plates," MED-Vol.2-1/MH-MH-Vol.3-1, Manufacturing Science and Engineering, ASME 1995, pp.3-14.

L. Cai and P. Sheng, 1996,"Analysis of laser evaporative and fusion cutting," Journal of Manufacturing Science and Engineering, May 1996, Vol.118, pp.225-234

Modest, M. F., 1996, "Three-dimensional, transient model for laser machining of ablating de-composing materials," Int. J. Heat Mass Transfer, Vol 39 (2), pp. 221-234.

Paek, U. C., and Gagliano, F. P., 1972, "Thermal analysis of laser drilling processes," IEEE J. of Quantum Electronics, Vol. QE-8, pp. 112-119.

Singh, R. K., and Narayan, J., 1990, "Pulsed-laser evaporation technique for deposition of thin films: physics and theoretical model," Phys. Rev. B 41(3), pp.8843-8859.