APPENDIX B 

 

DYNAMICS OF THE ABLATION PLUME

 

The ablation plume dynamics was coupled to the finite difference model, in order to estimate the pressure and attenuation of radiation in the plume.

 

B.1 INTRODUCTION

 

The morphology of a blast wave emanating from surfaces irradiated with excimer laser pulses may show distinct regions and a complex temporal evolution [1, 2]. As illustrated in Figure B.1.1 the shock front US1 precedes an ionisation front US2 and expands away from the surface as described by Sedov [3] for the blast wave accompanying an explosion. The radius R_sw of the shock wave front, which is assumed to be spherical, is

 

and	(B.1.1)       (B.1.2)

 

where E_blast is the energy deposited in the explosion, r₀ the ambient-gas density, t the time, and g  the ratio of specific heats. These expressions were derived assuming spherical symmetry, energy and momentum conservation, and an instantaneous, massless, point explosion. Equation (B.1.1) has been found to accurately represent the expansion of the shock front during excimer laser ablation, if E_blast is interpreted as the energy deposition up to the time t [2]. Using E_blast as an empirical parameter, Callies et al. [2] found that the fraction of the total pulse energy that is deposited in the shock wave by the end of the laser pulse depends strongly on the pulse energy, and may increase to about 80% of the incident pulse energy at high fluences. The discontinuity US2 is a region of luminosity that accompanies the absorption of laser radiation in an ionisation front. This front appears about halfway through the laser pulse and subsequently expands at the same speed as US1. US3 is the contact front between ablated material and the shock ambient gas. It starts as a planar expansion and then evolves after some time t₀ into a spherical expansion. After this time, the radius R_cf of the contact front and R_sw are related as follows

 

(B.1.3)

 

Equation (B.1.3) has been shown to accurately model R_cf during the laser pulse [2]. The lateral expansion giving rise to the discontinuity US4 can be attributed to the stagnation produced by plasma ignition (US5) near the surface. This yields a rapid radial surface expansion of vaporised material localised along the surface, which starts part way through the laser pulse and persists until the end pulse.

 

a)	b)
Figure B.1.1: Discontinuities observed in the blast wave emanating from a copper target irradiated in air at 248 nm. a) Laser induced density variation detected with the shadowgraph method 50 ns after the pulse (30 J/cm2). b) Schematic diagram of the descontinuities (from [2]).

 

B.2 MODELLING THE PLUME DYNAMICS

 

The flow of the laser-evaporated vapour may be described according to a simpler model developed by Knight [4]. When the laser fluence is higher than the ablation threshold but less than the threshold for explosive phase change the flow is schematically shown in Figure B.2.1a. The anisotropic velocity distribution of vapour particles escaping from the hot surface is transformed into an isotropic one by collisions among the vapour particles. This happens within a few mean-free paths (typically of the order of a few micron) from the surface, a region known as the Knudsen layer [5]. Some particles experience large angle collisions and are scattered back to the surface. In vacuum, the plume expands freely or adiabatically [6]; in ambient gas, the ablated material compresses the ambient gas, and forms a shock-wave, or expands in a mixed fashion in the case of low pressure of the background gas [6, 7]. When the laser fluence is higher than the threshold for explosive phase change, the superheated liquid turns into a mixture of liquid and vapour propagating into the air as shown in Figure B.2.1b. This mixture also forms a contact front with the air, and the compressed air propagates into the ambient, forming a shock. The exact plume behaviour in this situation is not considered in this work.

 

Figure B.2.1: Schematic representation of the ablation plume. a) Knight model [4], and b) Phase explosion model [8].

 

Using the conservation equations of mass, momentum and energy the velocity of the shock wave may be related to the surface temperature and pressure [9, 10]. The velocity of the shock wave (s_sw) is related to the pressure P_ca and temperature T_ca behind the shock by the Hugoniot equation [11] as

 

and	(B.2.1)     (B.2.2)

 

where M_sw is the Mach number of the shock wave, g_amb  is the adiabatic index of the ambient gas, and P₀ is the ambient pressure.  It is assumed that the pressure, temperature and velocity are uniform in the compressed air (ca). Therefore, at the contact front, P_cf, T_cf and v_cf may be assumed identical to those of the compressed air. The relations between the temperature and the pressure at the contact front and at the exit of the Knudsen layer, T_vapour and P_vapour are

 

and	(B.2.3)     (B.2.4)

 

where C_p is the specific heat of the vapour, P_abs is the absorbed laser power and r is the density. The vapour may be treated as an ideal gas [12], , and the expansion velocity of the vapour v_vapour equal to the local sound velocity , where g is the adiabatic index of the vapour and M the molecular or atomic mass. The vapour after the Knudsen layer is described differently according to the Mach number of the flow at the exit of the Knudsen layer, that is, M_vapour<1 or M_vapour=1 [4, 10]. Calculations have shown that for typical fluences used in excimer laser ablation, the Mach number at the exit of the Knudsen layer equals unity [13]. Anisimov [9] expressed the properties of the vapour as a function of the surface temperature T_surf, and taking g=5/3 and the M_vapour=1, found the results

 

Tvapour @ 0.65 Tsurf and

(B.2.5)   (B.2.6)

 

where  denotes the saturated vapour density at temperature T. This vapour is thus significantly cooler and less dense than the vapour in equilibrium with the surface. An analysis of the velocity redistribution in the Knudsen layer under the same assumptions showed that about 18% of all evaporating particles return to the surface [9]. The net evaporation rate is thus somehow smaller than the equilibrium value, resulting

 

.

(B.2.7)

 

The surface pressure may be calculated by considering that the dense phase is subject to recoil forces exerted by the evolving vapour. The actual surface pressure is found to be about half the saturated vapour pressure [9]

 

(B.2.8)

 

The total force on the irradiated body is the integral of P_surf over the irradiated area, while the momentum is the time integral of the force. For a pulse of duration t the total recoil momentum can be estimated from

 

(B.2.9)

 

where R_spot is the beam radius. Instead of using the conservation equations, blast wave theory may be used to derive the peak overpressure P_blast [14]

 

with and

(B.2.10)     (B.2.11)     (B.2.12)

 

where M_sw is the blast-wave’s Mach number, and C₀ is the sound speed in the ambient gas. The surface pressure due to the blast wave is maximum during the pulse [2], and is of the same order of the blast pressure [15]. Evaporating material is not taken into account in the Sedov theory, therefore the density in the material/vapour region is higher and the temperature is much lower than the predicted by the theory.

In the numerical calculations developed the positions of the blast wave and contact fronts are estimated using equations (B.1.1) and (B.1.3). It is assumed that the blast wave energy is delivered throughout the pulse duration, and is a fraction of the pulse energy as experimentally determined (Chapter 5). Hugoniot equation allows to calculate the pressure and temperature at the contact front T_ca [equation (B.2.1)], while the thermal model allows to estimate the surface temperature T_surf. The vapour temperature T_vapour is assumed as the average value of these temperatures. The density in the vapour r_vapour is estimated by considering a homogeneous distribution of the ablated material (as determined from the thermal model) from the surface to the contact front R_cf. The values r_vapour and vapour temperature T_vapour are used to calculated the vapour pressure P_vapour, and afterwards P_surf in accordance to equation (B.2.8). P_surf is used to calculate a “new” vaporisation temperature for the material, following Clausius-Clapeyron equation. Attenuation of incoming radiation in the plume by Mie absorption and scattering is estimated by using the vapour density r_vapour and cluster density values to calculate the corresponding attenuation coefficients, as shown in Chapter 5. The calculations were performed for every iteration (Dt»10^-11 s) during the pulse duration (t=60´10^-9 s).

The physical interpretation of the blast wave theory can be understood considering an elastic interaction model between the ablated particles and the ambient gas. Consequently, with increasing ambient pressures, the cross section for interaction between ablated particles and ambient particles increases. More collisions leading to momentum transfer take place during the same time and consequently the shock wave energy decreases more abruptly. On the other hand, with increasing atomic or molecular weight of the ambient gas, at each collision more energy is lost by particles in the shock wave, and therefore the shock wave energy is reduced faster. This decrease is more important for “light” species than for “heavy” ones [19], in agreement with the values of the transfer coefficient a_tc of kinetic energy in a elastic collision between a particle of mass m₁, belonging to the plume, and a background gas particle of mass m₂:

 

 

 

REFERENCES

 

1. A. D. Sappey and T. K. Gamble: Planar laser-induced fluorescence imaging of Cu atom and Cu₂ in a condensing laser-ablated copper plasma plume, J. Appl. Phys. 72 (11), 50955107 (1992)

2. G. Callies, P. Berger, and H. Hügel: Time-resolved observation of gas-dynamic discontinuities arising during excimer laser and their interpretation, J. Phys. D: Appl. Phys. 28, 794-806 (1995)

3. L. I. Sedov: Similarity and Dimensional Methods in Mechanics, (Cleaver Hume Press, London 1959)

4. C. J. Knight: AIAA J. Thermophysics Heat Transfer 17, 519-523 (1979)

5. M. v. Allmen: Laser-Beam Interactions with Materials: Physical Principles and Applications, (Springer-Verlag, Berlin Heidelberg 1987)

6. B. Angleraud, J. Aubreton, and A. Catherinot: Expansion of the ablation plume created by ultraviolet laser irradiation of various target materials, Eur. Phys. J. AP 5, 303-310 (1990)

7. A. Gusarov, A. Gnedovets, and I. Sumrov: Gas dynamics of laser ablation: influence of ambient atmosphere, J. Appl. Phys. 88 (7), 4352-4364 (2000)

8. X. Xu and K. Song: Phase change phenomena during high power laser-materials interaction, Mat. Sci. Eng. A 292, 162-168 (2000)

9. S. I. Anisimov: Vaporization of metal absorbing laser radiation, Sov. Phys. JETP 27 (1), 182-183 (1968)

10. C. L. Chan and J. Mazumder: One dimensional steady-state model for damage by vaporization and liquid expulsion due to laser-material interaction., J. Appl. Phys. 62 (11), 4579-4586 (1987)

11. H. W. Leipmann and A. Roshko: Elements of Gas Dynamics, (John Wiley & Sons, New York 1957)

12. V. A. Batanov, F. V. Bunkin, A. M. Prokhorov, and V. B. Fedorov: Evaporation of metallic targets caused by intense optical radiation, Sov. Phys. JETP 36 (2), 311-322 (1973)

13. K. H. Song and X. Xu: Mechanisms of absorption in pulsed excimer laser-induced plasma, Appl. Phys. A 65, 477-485 (1997)

14. Y. B. Zeldovich and Y. P. Raizer: Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, (Academic Press, New York 1966)

15. D. Bäuerle: Laser Processing and Chemistry, (Springer-Verlag, Berlin Heidelberg 2000)