APPENDIX A 

 

THERMAL MODEL

 

The finite-difference thermal model developed to investigate heating/ablation during irradiation with nanosecond duration laser pulses is presented. This model was used to investigate the heating behaviour of Al₂O₃, TiC and Al₂O₃-TiC by considering the fraction of the individual phases in the composite. It allowed to study the influence on heating of various material properties, namely the optical absorption coefficient and thermal conductivity.

 

A.1 INTRODUCTION

 

Most laser applications for materials processing purposes are based on the thermal action of laser radiation, causing changes in composition, structure, and properties of the material or leading to its ablation. The modification and ablation of solids under the effect of laser radiation proceed through phase transformations whose progress is governed mainly by heat and mass transfer processes. The physical and mathematical models of such processes are frequently constructed on the basis of the heat conduction and diffusion theory.

For the macroscopic heat conduction theory to be applicable, it is necessary that the system is in a state of local equilibrium, which in condensed media takes a time of about  to be established [1]. If the duration t of the processes one is interested in is , then the system may be described in statistical terms, in order to determine the average velocity, density, and energy of particles; that is, the notions of temperature, phase, and other thermodynamic parameters may be used [1]. The classical heat conduction theory postulates the validity of the Fourier heat transfer law and uses the parabolic differential equation [2]

(A.1.1)

 

where G is the heat source. If heat propagates in the direction z only, and the thermophysical constants C_p, K and r are independent of the temperature, equation (A.1.1) assumes the form

 

(A.1.2)

Equations (A.1.1) and (A.1.2) presume an infinite propagation velocity v* of thermal perturbations. When the intensity of the heat source is high, the finiteness of v* may be important for fast processes, and the differential heat transfer equation will then have the form [2]

 

(A.1.3)

 

In this expression,  is the thermal diffusivity and  is the relaxation time, which ranges between 10^-11 and 10^-12 s in metals, and between 10^-10 and 10^-11 s in porous solids. Therefore, if the characteristic time of laser-induced processes exceeds about 10^-9 s, the traditional equations (A.1.1) and (A.1.2) can be used, and the analysis of the absorption of laser energy and its transfer to the lattice can be made by considering a laser heat source and uniform condensed phase temperature. Under these conditions the electromagnetic energy from the laser light can be viewed as being instantaneously converted into heat at the location of absorption.

The modelling of heat transfer in the substrate irradiated by a laser pulse of short duration is typically performed using the Fourier heat diffusion model. Instead of this model, the hyperbolic heat equation [3], the vibrational cooling model [4,5], and the equation of phonon radiative transfer [6] based on Boltzmann transport theory have been developed.

 

A.2 ONE-DIMENSIONAL MODEL

 

A finite-difference model was developed to study heating upon irradiation with nanosecond duration pulses at excimer laser wavelengths. The model is based on the work developed by Wagner [3]. Since the cross section of the incident laser beam (frequently above 50 mm) is significantly larger than the typical optical penetration depth (<2 mm), a one-dimensional approach may be applied to the situation under study. The one-dimensional model was applied to Al₂O₃, TiC and a hypothetical material whose properties were estimated considering the weight fraction of Al₂O₃ and TiC in the Al₂O₃-TiC composite. The target is divided in several elements, and heating of any element is accomplished by bulk absorption of laser energy that has been transmitted through elements near the surface. The transient temperature behaviour of each element in a slab is calculated by considering its specific heat and conduction to neighbouring elements. In a first approximation heating is supposed to occur at atmospheric pressure, and temperatures above the vaporisation temperature are not supported. At the vaporisation temperature, an element absorbs the latent heat (which is supplied by the laser energy) at constant temperature and pressure.

It is assumed that the material is irradiated with a laser beam of uniform power density I₀(t) (W/cm²), and has an absorption coefficient a such that the intensity I(z,t) (W/cm²) at any depth z from the surface of the material is given by I(z,t)=(1-R)I₀(t)exp^-az, where R is the surface reflectivity. Under these conditions, the absorption of laser energy is equivalent to heating the material with a non-uniform heat source w(z,t) (W/cm³), where w(z,t)=-dI(z,t)/dz=(1-R) I₀(t)aexp^-az. The temporal variation of laser intensity I₀(t) is approximated by considering a triangular pulse with a duration at its base of t_l = 60 ns, and peak intensity at t_p = 10 ns, in agreement with the behaviour of standard nanosecond excimer lasers, resulting

 

(A.2.1)

 

For the purposes of numerical analysis the material is divided into n nodes of width d. Each node undergoes the following heating process: 1) It absorbs heat at constant specific heat C_p until it reaches the melting temperature T_m. 2) Then, absorbs heat at constant temperature until it has absorbed the heat of fusion H_f. 3) It is again heated at constant specific heat to the vaporisation temperature. 4) Finally, it absorbs the latent heat of vaporisation H_v at constant temperature and is vaporised. From an energy balance on the ith node the temperature rise DT_i of that node during the time Dt can be found

 

(A.2.2)

 

where Q_i (J/cm²) is the heat input to the ith node. The heat input Q_i depends upon the heat-source strength w(z_i, t) and upon conduction from adjacent nodes

 

(A.2.3)

 

where K is the thermal conductivity and z_i=(i-1)d. Combining equations (A.2.2) and (A.2.3) gives a result which allows numerical calculation of the transient temperature behaviour in the ith node:

 

(A.2.4)

 

When a node reaches the melting temperature it absorbs the heat of fusion at constant temperature. The technique to handle this situation numerically is to define an auxiliary variable P_i. When the ith node reaches the melting temperature, P_i accumulates the calculated value of the heat input C_pDT_i for enough iterations to account for the heat of fusion H_f. When  during melting, the accumulation process is complete and the node is again heated according to equation (A.2.4). If a node reaches the vaporisation temperature it must absorb the heat of vaporisation H_v, and the accumulation process is resumed. Whenever P_i=H_v+H_f, the node vaporises and vanishes. Then a new surface exists on the slab and the definition of z_i must be revised. If a total of m surface nodes have been vaporised then z_i=(i-1-m)d. The computation of C_pDT for the first and last nodes is slightly different, since conduction can occur to only one node, modifying equation (A.2.4) as follows

 

, for i = m+1   , for i = m+1, m+1, ..., n-1   , for i = n

(A.2.5)

 

In summary, equation (A.2.5) is used in conjunction with the auxiliary variable P_i to calculate the transient temperature behaviour of the material irradiated by the laser beam. The dependence on temperature of the thermal properties is considered by using a linear interpolation of the values shown in Table A.2.1. The variation with temperature of the optical properties is not expected to be significant with for UV wavelengths [4], and it is not considered. The calculations were performed considering the width d of the individuals nodes as 10 and 1 nm. For convergence to be achieved, these values require that the temperature along the depth of the material is calculated every ~10^-11 and 10^-13 s, respectively. Since these different time intervals did not lead to significant variation on the results, most calculations were performed considering a 10 nm node width, in order to minimise the computing time for a pulse duration of about 60´10^-9 s. 

In order to consider temperatures in the material above the vaporisation temperature at atmospheric pressure, the Clausius-Clapeyron equation was used and the pressure and temperature in the plume estimated as shown in Appendix B. A two-dimensional finite-difference model was also developed, and consists in an extension of the simpler one-dimensional version. This model allowed to consider TiC grains in an Al₂O₃ matrix, and the influence of the surface topography on heating.

 

 

 

 

Table A.2.1a: Parameters used in numeric calculations. Properties of Al₂O₃ and TiC collected from CRC Handbook of Chemistry and Physics [5], TAPP [6] where not indicated otherwise.

 

 

Table A.2.2b: Temperature dependence of physical properties [6].

Temperature dependent properties for TiC
	298 K	1819 K	3300 K	liquid
Specific heat Cp (J/kg.K)	563	960	1230	1048
Density of mass r (kg/m3)	4930	4804	4682	4400
Thermal conductivity K (W/mK) [9]	9	40	50	50
Temperature dependent properties for Alumina
	298 K	1312 K	2327 K	liquid
Specific heat Cp (J/kg.K)	774	1275	1363	1888
Density of mass r (kg/m3)	3990	3924	3860	3200
Thermal conductivity K (W/mK) [9]	36	6	5	5

 

REFERENCES

 

1. E. N. Sobol: Phase Transformations and Ablation in Laser-Treated Solids, (John Wiley & Sons, New York 1995)

2. H. S. Carslaw and J. C. Jaeger: Conduction of heat in solids, (Oxford University Press, Oxford 1959)

3. R. E. Wagner: Laser drilling mechanics, J. Appl. Phys. 45 (10), 4631-4637 (1974)

4. W. W. Duley: UV lasers: Effects and Applications in Materials Science, (Cambridge University Press, 1996)

5. D. R. Lide: CRC Handbook of Chemistry and Physics, (CRC Press, 1996)

6. TAPP: Thermochemical and Physical Properties (ES Microware, 1991)

7. J. Pflüger and J. Fink: In Handbook of Optical Constants of Solids II (Academic Press, 1991), pp. 303-311

8. V. N. Tokarev, W. Marine, C. Prat, and M. Sentis: Clean processing of polymers and smoothing of ceramics by pulsed laser melting, J. Appl. Phys. 77 (9), 4714-4723 (1995)

9. M. V. Swain: Structure and properties of ceramics, R.W. Cahn, P. Haasen, and E.J. Kramer (VCH, New York 1994)