Sequences of Sub-Microsecond Laser Pulses for Material Processing: Modeling of Coupled Gas Dynamics and Heat Transfer

Featured Application: Material processing by pulsed lasers can su ﬀ er from plasma ignition, which reduces e ﬃ ciency. This work compares the conditions at single- and multipulse laser processing and proposes a modeling tool to optimize the sequences of pulses. Abstract: Multipulse laser processing of materials is promising because of the additional possibilities to control the thickness of the treated and the heat-a ﬀ ected zones and the energy e ﬃ ciency. To study the physics of mutual interaction of pulses at high repetition rate, a model is proposed where heat transfer in the target and gas-dynamics of vapor and ambient gas are coupled by the gas-dynamic boundary conditions of evaporation / condensation. Numerical calculations are accomplished for a substrate of an austenitic steel subjected to a 300 ns single pulse of CO 2 laser and a sequence of the similar pulses with lower intensity and 10 µ s inter-pulse separation assuring approximately the same thermal impact on the target. It is revealed that the pulses of the sequence interact due to heat accumulation in the target but they cannot interact through the gas phase. Evaporation is considerably more intensive at the single-pulse processing. The vapor is slightly ionized and absorbs the infrared laser radiation by inverse bremsstrahlung. The estimated absorption coe ﬃ cient and the optical thickness of the vapor domain are considerably greater for the single-pulse regime. The absorption initiates optical breakdown and the ignition of plasma shielding the target from laser radiation. The multipulse laser processing can be applied to avoid plasma ignition.


Introduction
Pulsed laser processing of materials has some advantages over continuum-wave processing because it can offer thinner processed and heat-affected zones and higher surface temperatures. The serious drawback of laser pulses with durations ranging from nanoseconds to microseconds is the ignition of plasma. The plasma shields the treated surface from the laser beam. Thus, the efficiency of laser treatment decreases with the pulse energy. High repetition-rate sequences of pulses is a well-known solution to increase the efficiency [1][2][3][4].
The computational model developed by Vora et al. [2] to understand the influence of multiple millisecond laser pulses on the surface finish is in agreement with experimentally observed surface roughness [2]. Ranjbar et al. [3] numerically simulated the dynamics of vapor plumes induced by nanosecond multi-pulse laser irradiation in the burst mode. They showed that the multi-pulse mode can considerably decrease the plasma ignition threshold and increase the efficiency of ablation. the kinetic description of the gas phase was used in the framework of DSMC [12,13]. Such an approach has been successfully applied to nanosecond laser ablation up to now [3]. However, the DSMC computation becomes extremely time consuming at small Knudsen numbers. This is why it is not applied in the recent models for sub-microsecond [4] and millisecond [3] laser pulses.
The present work implements a rigorous but less time-consuming computation model where the interface region of the Knudsen layer is considered explicitly [15]. It concerns modeling the sequences of CO 2 -laser pulses with 10.7 µm wavelength, the duration of 300 ns, and inter-pulse separation of 10 µs, which are promising for high-efficiency surface heat treatment [28]. Estimation of the ionization degree [27] and especially modeling the optical breakdown [4,16] require strong assumptions. To obtain conclusions that are more general, we restrict our consideration to qualitative effects of laser-plasma interaction. The calculation results obtained are rigorous before plasma ignition only. Note that both experiments and modeling [4,16] indicate that plasma ignition instantly interrupts the laser beam. It should be avoided at laser processing. The goals of the present study are to estimate the threshold of evaporation, the parameters of the vapor important for plasma ignition, and the physical mechanisms governing the mutual influence of the pulses in the multipulse sequences. The fraction of laser energy absorbed in the gas phase depends on its optical thickness. This is why the probability of plasma ignition is assessed by the optical thickness.

Model
In this model, heat transfer in the target and the dynamics of the ambient gas and the vapor are coupled through the boundary conditions at the interface. Only the central region of the laser spot is considered where the laser energy flux is supposed to be uniform. The thermal and the gas-dynamic responses are also uniform over this domain. Thus, the problem is essentially one-dimensional, at least at the initial stage of laser-material interaction. The position of the evaporating surface is not fixed, its recession velocity u s being consistent with gas-dynamic and temperature fields. Recession velocity due to laser evaporation is much smaller than the characteristic gas-dynamic velocities [15]. Therefore, it is neglected in the gas-dynamic problem considered in half-space z > 0 of the frame where the positive direction of axis Z is the external normal to the target surface approximated by plane z = 0 and the origin is fixed at the surface (see Figure 1). By contrast, the recession can influence heat transfer in the target [15] considered in half-space z < 0. The mathematical model used in this work is the thermal model of laser ablation [14,15], which was initially applied for single nanosecond pulses. It consists of the heat transfer equation in the target and the gas-dynamic equations for vapor and ambient gas. The heat transfer and gas dynamics are coupled by the gas-dynamic boundary conditions of evaporation/condensation [26].

Governing Equations
Heat conduction is supposed to be the principal mechanism of heat transfer in the target and is described by the heat equation in enthalpy form [14,15].
where h is the enthalpy per unit volume, t the time, λ the thermal conductivity, and T the temperature. The second term in the left hand side of Equation (1) is responsible for advection in the frame-moving relative medium. In the case of net evaporation, the interface recesses, the medium of the target moves in the positive direction of axis Z, and u s > 0. In the case of net condensation, the interface rises, the medium of the target moves in the negative direction of axis Z, and u s < 0. The volumetric specific heats in solid c s and liquid c l phases are approximately taken as constants. Then, the thermal equation of state [14], relates the enthalpy and the temperature, where T m is the melting point and q m the latent heat of melting per unit volume. Gas-dynamic equations of continuity (3) and (4), momentum (5) and energy (6) [15], are applied at z > 0, where n v and n g are the number densities of vapor and ambient gas, respectively, u the gas velocity, ρ the gas density, p the pressure and E the internal energy per unit volume. In the gas mixture [15], ρ = m v n v + m g n g , where m v and m g are the molecular masses and γ v and γ g the adiabatic exponents of vapor and ambient gas, respectively. Mass balance at the interface, z = 0, is [15] n t u s = n v u, where n t is the number density in the target. If laser energy is absorbed at the target surface with the heat effect per unit time per unit area equal to absorbed laser energy flux K a , the energy balance at where T s is the temperature of the target surface and U the binding energy of target atoms at T = 0. Note that gas temperature at the interface T(0) is necessary to calculate the right hand side of Equation (11). Temperature T(0) is generally different from T s because of a temperature jump across the Knudsen layer formed near the interface at strong evaporation or condensation. Binding energy U is the principal part of the latent heat of evaporation q v . In addition to Equations (10) and (11), two boundary conditions at z = 0 are necessary at evaporation and only one boundary condition is applied at condensation [26]. Gas-dynamic boundary conditions of evaporation/condensation relate the parameters of target surface with the parameters of vapor formed at the outer edge of the Knudsen layer. The target parameters are its surface temperature T s and saturated vapor pressure p s at T s . The saturated vapor pressure is estimated by integrating the Clausius-Clapeyron equation in assumption that the latent heat of evaporation q v is independent of temperature [14] where T b is the boiling temperature at atmospheric pressure p a . The thickness of the Knudsen layer at strong evaporation/condensation is about few mean free paths [26]. It is neglected in this work. Therefore, gas parameters, such as temperature T, pressure p and Mach number M are taken at z = 0. For example, the Mach number is defined as [15] At evaporation M > 0 while at condensation M < 0. Boundary conditions of strong evaporation/condensation were obtained by gas-kinetic calculations of the Knudsen layer [20][21][22][23][24][25][26]. Analytical boundary conditions used in this work approximate the above cited rigorous calculations under the conditions typical for laser processing [15]. Evaporation and condensation are distinguished by comparing gas pressure p and saturated vapor pressure p s . At evaporation, p < p s , the two boundary conditions are [15]: where p 0 = 0.20742 is the pressure ratio and T 0 = 0.6437 the temperature ratio at evaporation with M = 1. Note that M > 1 is impossible at evaporation [20][21][22][23][24][25][26]. At condensation, p > p s , the boundary condition is [15] − where parameters α and I 0 are functions of temperature ratio T/T s specified in [15].

Computations
Partial differential Equations (1), (3)- (6) with boundary conditions at the interface given by Equations (10), (11), (14)- (16) are solved in computation domains shown in Figure 1 by the numerical method of finite volumes on uniform grids. The parameters of undisturbed ambient gas are imposed on the top boundary of the gas-phase domain. Ambient temperature T a is the boundary condition at the bottom of the target domain. When the shock wave approaches to the top of the gas-phase domain or the heat-affected zone approaches to the bottom of the target domain, the corresponding computation domain is enlarged, the grid is rearranged, and the discrete fields from the previous grid are interpolated to obtain new discrete fields on the new grid. Thus, the computation domains are always greater than the disturbed zones, and the influence of the external boundaries is excluded. Differential Equations (1), (3)-(6) are approximated by second order in space-and first order in time-explicit finite difference equations. A second-order Godunov scheme [29] is used for gas-dynamic Equations (3)-(6). Central differences are applied to the spatial derivatives of heat Equation (1).
The present work uses original software for the thermal model of laser ablation, which was developed earlier [15]. The details of the numerical method and the calculation algorithm are given elsewhere [15].

Results
Laser treatment of austenitic steel X18H10T by pulsed CO 2 laser in an air atmosphere is modeled. The parameters of the target and the ambient gas are listed in Tables 1 and 2, respectively. Binding energy U from Equation (11) is approximated by latent heat of evaporation q v . The temporal profile of the laser pulse is specified by a triangle with the increasing front of τ f = 50 ns and the decreasing back of τ b = 250 ns (see Figure 2). The maximum of intensity I max is a variable parameter. It is attained at 50 ns after the pulse beginning. To exclude the uncertainty due to the absorptance of the target, the absorbed part of laser energy K a is specified as the model parameter. In the energy balance, Equation (11), it is explicitly defined as the following function of time:  [30].  Density at the ambient conditions, ρ 0 (kg/m 3 ) 1.185 [30] Atomic mass, m g , (a.m.u.) 29 [30] Adiabatic exponent, µ g 7/5 [30] The pulse parameters are listed in Table 3. Figures 2-4 show the effect of a single pulse with the maximum absorbed intensity I max = 15 MW/cm 2 . Figure 2 shows the temporal profile of the laser pulse specified by the function of absorbed laser power K a versus time t. Figure 3 shows temperature fields in the target calculated for this pulse. Surface temperature attains the boiling point T b and becomes greater than T b . Therefore, an intensive vapor flow is formed along with a compressible flow of ambient air. Figure 4 shows gas-dynamic fields. Notice a shock front separating undisturbed and disturbed domains of air and a contact jump between air and vapor. The shock front is very close to the boundary of the computation domain for t = 0.2, 0.3, and 1 µs. Those are the instances just before the enlargement of the domain and the rearrangement of the computation grid. The shock front never touch the right boundary. At t = 0.1 and 0.2 µs (see Figure 4), gas-dynamic velocity u is positive at the interface z = 0 indicating net evaporation but at the end of the laser pulse at t = 0.3 µs, u(0) becomes negative indicating net condensation. Figure 2 shows the dynamics of the surface temperature T s and the heat loss to evaporation calculated from the right hand side of Equation (11). One can see that the evaporation loss is much smaller than the absorbed laser energy for this pulse and the surface temperature can be considerably greater than the boiling point T b = 3200 K.  Table 3. Parameters of the laser pulse.

Duration (ns) 300
Inter-pulse separation (µs) 10 Absorbed intensity at the maximum, I max (MW/cm 2 ) 10 . . . 15 A single pulse with smaller intensity I max = 10 MW/cm 2 cannot heat the target up to the boiling point (see the dotted line in Figure 5). However, the heat of the pulse is accumulated in the surface layer of the target. Figure 6 shows that at the moment of t = 10 µs after the pulse, the surface temperature is still about 600 K. The second pulse of the sequence starting at t = 10 µs, interacts with the already heated target. Therefore, it results in higher surface temperature (dashed line in Figure 5). The surface temperature becomes greater than the boiling point and evaporation starts. At the beginning of the third pulse at t = 20 µs, the surface temperature attains 800 K (see Figure 6a), the temperature after the third pulse is higher (full line in Figure 5), and evaporation should be more intensive. Figure 6b shows the evolution of the temperature field in the target after the third pulse and Figure 7 presents the gas-dynamic fields. The gas-dynamic fields are qualitatively similar to the fields shown in Figure 4 for a single pulse with I max = 15 MW/cm 2 . The maximum vapor pressure in Figure 4 is about 5 atm. while it is around 2 atm. In Figure 7. Therefore, one can conclude that evaporation is less intensive at I max = 10 MW/cm 2 , even after the third pulse.   Evaporation/condensation at the target surface is characterized by such parameters of the Knudsen layer as pressure ratio p/p s , temperature ratio T/T s , and Mach number M related by the boundary conditions of evaporation/condensation, Equations (14)- (16). Figure 8 shows the evolution of these parameters at the studied regimes of laser processing. The effect of laser radiation on the target is characterized by the evolution of the temperature field shown in Figures 3 and 6 along with the melt depth in the target shown in Figure 9.

Discussion
The modeling indicates that at inter-pulse separation 10 µs, the heat transferred to the target by a laser pulse is not completely dissipated when the next pulse of the sequence starts. Thus, thermal energy is accumulated in a near-surface layer of the target. It is illustrated by Figure 5 where the temperature curve of each pulse is above the temperature curve of the previous pulse. The intensity of evaporation by the pulses of a sequence increases because of increasing surface temperature.
In the modeled regimes of single and multipulse laser processing, the vapor velocity near the target surface becomes negative even before the termination of the 300 ns laser pulse (see Figures 4 and 7). Thus, back vapor condensation on the target surface starts. According to the model, all the evaporated material condenses back. The period of time necessary for full back condensation depends on the parameters of laser processing. The drop of the absolute value of Mach number |M| down to zero in Figure 8 indicates the instant of complete back condensation. For single pulse with I max = 15 MW/cm 2 (see Figure 8b), it is t = 1900 ns. At I max = 10 MW/cm 2 , the duration of back condensation is considerably less. For the second pulse (see Figure 8c), t = 315 ns. For the third pulse (see Figure 8d), t = 415 ns. In any case, back condensation finishes well before the beginning of the next laser pulse at t = 10 µs. This is why the laser radiation of the next pulse cannot interact with the vapor formed by the previous pulse. Figure 8 shows that Mach number M approaches its theoretical limit for evaporation M = 1 [20-22] but does not attain it. This is because of a relatively low laser intensity. Calculations by the similar model at higher intensities [15] indicated that M = 1 can be attained. At evaporation, temperature ratio T/T s and pressure ratio p/p s are functions of M given by Equations (14) and (15). Note that the temperature ratio attains its minimum when evaporation is the most intensive (see Figure 8). At the stage of condensation, M tends to −1 (see Figure 8). Note that supersonic condensation with |M| > 1 is theoretically possible [24,25]. However, it is not observed in the present work as well as in the previous one [15]. For the calculated examples, temperature ratio at condensation is in the interval between 0.4 and 1 (see Figure 8). This means that the vapor is colder than the target. Such an unusual ratio is explained by adiabatic expansion of vapor where its temperature rapidly decreases as a power function of the density. The temperature of the target surface is controlled by heat conduction and decreases more slowly. Even if the vapor is colder, its pressure is still greater than the saturated vapor pressure at the temperature of the target surface. This is why condensation continues. Once the Mach number of the condensing vapor flow attains −1, the value of M is around −1 till the complete condensation of the vapor (see Figure 8). Similar behavior of the condensing vapor flow was observed at modeling for much shorter nanosecond pulses [15]. Figure 9 shows the calculated dynamics of the melting/solidification front. The maximum melt depth estimates the thickness of the surface layer with the modified structure. This value appears to be approximately the same for the single laser pulse with I max = 15 MW/cm 2 and for the sequence of three weaker pulses with I max = 10 MW/cm 2 (see Figure 9). It is slightly greater than one micron. However, the parameters of vapor are quite different. Table 4 compares the parameters at 200 ns after the beginning of the single pulse with I max = 15 MW/cm 2 (t = 0.2 µs in Figure 4) and the third pulse of the sequence with I max = 10 MW/cm 2 (t = 20.2 µs in Figure 7). In both cases, the vapor temperature is about 3000 K. The vapor pressure at 15 MW/cm 2 is four times the pressure at 10 MW/cm 2 . The thickness of the vapor domain at 15 MW/cm 2 is four times its thickness at 10 MW/cm 2 . In summary, at the miltipulse laser processing, vapor is considerably less dense and occupies a considerably thinner domain. Metallic vapor is expected to be slightly ionized at the temperature of 3000 K. The source of charges is thermoelectron and thermoion emission from the target surface [27]. Modeling [27] revealed the Knudsen layer and the sheath fused together at the interface between an evaporating surface and slightly ionized vapor. It was shown that the electronic temperature is around the surface temperature and the ionization degree is around the equilibrium one at the electronic temperature [27]. The last row of Table 4 shows the calculated surface temperature estimating the electronic temperature of the vapor. One can see that the electronic temperature is greater for the single pulse with I max = 15 MW/cm 2 . According to the Saha Equation [31], the ionization degree sharply grows with the electronic temperature. Therefore, the ionization degree can be considerably greater at 15 MW/cm 2 . The principal mechanism of the absorption of infrared radiation in ionized gas is inverse bremsstrahlung [31]. The absorption coefficient is proportional to the concentration of electrons [31].
The probability of optical breakdown in the vapor can be assessed from the optical thickness, which is roughly the vapor layer thickness (optical path) multiplied by the absorption coefficient of the vapor. Compare, for example, the distributions of vapor density at the end of the single laser pulse (see Figure 4, t = 0.3 µs) and at the end of the third pulse in the sequence (see Figure 7, t = 20.3 µs). The thickness of the vapor domain is about 140 µm in the former case and only 30 µm in the latter case. Thus, the optical path through vapor considerably reduces at the multipulse laser processing. One can see from these figures that the vapor density is around the density of the ambient gas in the former case and approximately a quarter of the ambient gas density in the latter case. The electron temperature and the ionization degree in the vapor are lower at the multipulse laser processing (see above). The electron density is the vapor density multiplied by the ionization degree. Therefore, the electron density along with the absorption coefficient considerably reduces at the multipulse laser processing. Finally, the optical thickness of the vapor considerably reduces too. The absorption initiates optical breakdown and the ignition of plasma shielding the target from laser radiation. Thus, the calculations indicate that at the miltipulse laser processing, the vapor parameters are considerably less favorable for plasma ignition while the thermal impact on the target is essentially the same. This is why the multipulse processing can be applied to avoid plasma ignition.
The present work proposes a modeling tool, which gives tendencies and can be a guideline in experiments. To the best knowledge of the authors, the present work is the first where gas-kinetic processes are taken into account and analyzed for sub-microsecond laser pulses. The two examples considered show the contribution of different physical processes. Based on the achieved understanding of the competing processes, one can qualitatively estimate the influence of the laser parameters. For the given laser setup, only two parameters are variable: the pulse energy and the repetition rate. The modeling indicates that increasing the pulse energy sharply increases the optical thickness of the vapor, which results in optical breakdown. This is why increasing the pulse energy is not desirable. On the other hand, increasing the repetition rate above the 100 kHz considered here (corresponds to the 10 µs inter-pulse separation) will increase heat accumulation in the target at a much lower value of the optical thickness. Thus, the thermal impact on the target will increase while the probability of plasma ignition will be still low. Therefore, increasing the repetition rate above 100 kHz seems to be promising in order to increase the efficiency of the laser processing.

•
Heat transfer in the target and gas dynamics of vapor and ambient gas coupled by gas-dynamic boundary conditions of evaporation/condensation are calculated for a single laser pulse with the maximum absorbed power density I max = 15 MW/cm 2 and a multipulse laser processing with I max = 10 MW/cm 2 .

•
The calculations indicate that the thermal impact on the target is essentially the same for the single pulse and the multipulse sequences. However, the vapor parameters are very different. Generally, evaporation is considerably more intensive at the single-pulse processing. • Thermal impact on the target and evaporation become more intensive with an increase in the number of pulses in the sequence because of heat accumulation in the target.

•
In the considered regimes, vapor formed during a laser pulse condenses back on the target surface during the time interval between consecutive pulses. Therefore, the pulses of the sequence can not interact through the gas phase.

•
The modeling indicates that the absorption coefficient and the total optical thickness are considerably greater in single-pulse laser processing. The absorption initiates optical breakdown and the ignition of plasma shielding the target from laser radiation. Therefore, multipulse laser processing can be applied to avoid plasma ignition.