Effect of the spot size on ionization and degree of plasma shielding in plumes induced by irradiation of a copper target by multiple short laser pulses

Abstract

The plasma plume expansion into argon background gas at atmospheric pressure induced by irradiation of a copper target with a burst of three short laser pulses at 266 nm wavelength is studied numerically for the laser spot diameters ranging from 20 μm to 500 µm. The computational model includes a thermal model of the irradiated target and a kinetic model of plume expansion. The kinetic model is implemented in the form of the direct simulation Monte Carlo method that is redesigned to account for ionization and absorption of laser radiation in the plume. The irradiation conditions are chosen to do not induce ionization and absorption during the first pulse in the burst independently of the laser spot size. During the second pulse, the ionization is initiated in the vicinity of the irradiated target behind the shock wave that is generated during that pulse and propagates through the vapor plume created by the preceding pulse. The simulations show that the degree of ionization and plasma shielding during the second and subsequent pulses strongly increases with increasing the laser spot size. It is explained by different  rates of expansion between pulses in the plumes generated at various spot sizes. At a relatively small spot size, the rapid drop of density and temperature in the plume induced by the first pulse can preclude plasma ignition during the second and further pulses. These results suggest that the use of lasers with the spot sizes that are in the order of tens of micrometers can be favorable for mitigating the effect of plasma shielding in multi-pulse laser ablation when the plumes induced by individual laser pulses strongly interact with each other.

Appendices

Appendix 1. Direct simulation Monte Carlo method for plasma plume flows

The extension of the DSMC method for simulations of laser-induced plasma plume flows is developed to capture the most important effects of ionization and radiation absorption on the energy budget. The approach developed for simulations of plasma flows is based on the assumption of locally neutral plasma, which allows one to do not consider the electrons as simulated particles. Instead, one can assume that electrons move together with their parent ions and are characterized only by the electron temperature $T_{\text{e}}$. Moreover, in our approach, we use a method of “lumped” particles, when a single simulated particle in the DSMC method represents neutral atoms and ions of the same species with various charges $z$ ($z = 0, \ldots ,Z_{\text{max} }$, where $eZ_{\text{max} }$ is the maximum ion charge considered in simulations, $e$ is the charge of an electron). This is the extension of the approach adopted in the classical DSMC method, where every simulated particle represents $W$ identical particles in the real gas flow ($W$ is referred to as the statistical weight; usually $W \gg 1$). Then, in plasma flows, one can assume that a single simulated particle $i$ of species $s_{i}$ ($s_{i} = 1$ for vapor and $s_{i} = 2$ for the background gas) simultaneously represents $WX_{i}^{z}$ ions with charges $z$, where $X_{i}^{z}$ is the molar fraction of corresponding ions $(z = 0, \ldots ,Z_{\text{max} }$; $\sum\nolimits_{z} {X_{i}^{z} } = 1$). Thus, the current state of any simulated particle is defined by its translational velocity ${\mathbf{v}}_{i}$, molar fractions of ions $X_{i}^{z}$, and temperature of electrons $T_{{{\text{e}},i}}$ associated with this particle. In the course of a simulation, values of $X_{i}^{z}$ and $T_{{{\text{e}},i}}$ are updated due to ionization and recombination process, as well as radiation absorption. This approach can be straightforwardly implemented in virtually any DSMC code.

In order to implement collisions between heavy particles, as well as ionization and absorption processes, the computational domain is divided into a mesh of cells. If a cell of volume $V$ contains $N$ simulated particles, then the number density $n^{z}$ of ions with charge $z$, number density of electrons $n_{\text{e}}$, macroscopic velocity ${\mathbf{u}}$, electron energy density $E_{\text{e}}$, thermal energy density of heavy particles $E_{\text{a}}$, and total thermal energy density $E$, which includes the energy of ionization, in this cell can be calculated as follows

$$n^{z} = \frac{W}{V} \mathop \sum \limits_{i = 1}^{N} X_{i}^{z} ,$$

(4a)

$$n_{\text{e}} = \frac{W}{V} \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{z = 1}^{{Z_{\text{max} } }} zX_{i}^{z} ,$$

(4b)

$${\mathbf{u}} = \mathop \sum \limits_{i = 1}^{N} m_{i} {\mathbf{v}}_{i} /\mathop \sum \limits_{i = 1}^{N} m_{i} ,$$

(4c)

$$E_{\text{e}} = \frac{W}{V} \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{z = 1}^{{Z_{\text{max} } }} \frac{3}{2}zX_{i}^{z} k_{\text{B}} T_{{{\text{e}},i}} ,$$

(4d)

$$E_{\text{a}} = \frac{W}{V}\mathop \sum \limits_{i = 1}^{N} \frac{{m_{i} \left( {{\mathbf{v}}_{i} - {\mathbf{u}}} \right)^{2} }}{2},$$

(4e)

$$E = E_{\text{a}} + E_{\text{e}} + \frac{W}{V}\mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{z = 1}^{{Z_{\text{max} } }} X_{i}^{z} \mathop \sum \limits_{m = 1}^{z} \text{IP}_{{s_{i} }}^{m} ,$$

(4f)

where $m_{i}$ is the mass particle $i$, ${\text{IP}}_{s}^{z}$ is the $z$-th ionization energy of species $s$, and $k_{\text{B}}$ is the Boltzmann constant. Each time step of the computational algorithm includes four major sub-steps: (1) Motion of particles, which leads to the redistribution of particles between cells, (2) binary collisions between heavy particles in cells, (3) absorption of laser radiation in cells, (4) ionization and recombination of particles in cells.

During the motion step, we account for the effect of electron pressure that is important in laser-induced plasma plumes. This effect is usually taken into account in hydrodynamic models of plasma plume expansion, e.g., [12,13,14,15]. For this purpose, we calculate the averaged electron temperature $T_{\text{e}} = E_{\text{e}} /\left( {\left( {3/2} \right)n_{\text{e}} k_{\text{B}} } \right)$ and pressure $p_{\text{e}} = n_{\text{e}} k_{\text{B}} T_{\text{e}}$ in each cells and then determine numerically the gradient of the electron pressure $\nabla p_{\text{e}}$. The equation of motion for every simulated particle in the form

$$m_{i} \frac{{{\text{d}}\mathbf{v}_{i} }}{{{\text{d}}t}} = - \frac{1}{{n_{\text{a}} }}\nabla p_{\text{e}} ,$$

(5)

where $n_{\text{a}} = WN/V$ is the total number density of heavy particles, is solved numerically with the second-order Runge–Kutta method. Binary collision between heavy particles is implemented based on the approach used in simulations of neutral gas flows [26].

The developed computational framework for simulations of plasma flows based on the modified DSMC method can be used for simulations of non-equilibrium plasma flows, where the local degrees of ionization are determined by the kinetic rates of ionization and recombination, e.g., [12], and the election temperature differs from the temperature of heavy particles $T_{\text{a}} = E_{\text{a}} /\left( {\left( {3/2} \right)n_{\text{a}} k_{\text{B}} } \right)$. In the present paper, however, the developed approach is used assuming the equilibrium degrees of ionization predicted by the Saha equations and $T_{\text{a}} = T_{\text{e}}$.

The propagation of laser radiation through the plasma plume is described by Beer’s law in the form ${\text{d}}I/{\text{d}}x = \alpha I$, where $I = I\left( {x,r,t} \right)$ is the intensity of laser radiation in a point with coordinates $\left( {x,r} \right)$ at time $t$ ($x$ is the coordinate counted along the laser beam from the irradiated surface and $r$ is the radial coordinate counted from the laser beam axis along the target surface), and $\alpha$ is the plasma linear absorption coefficient. In the present work, the absorption coefficient is calculated in each cell of the computational mesh based on the model presented in Appendix 2. The equation of radiation transfer is solved with the initial condition $I\left( {x,r,t} \right) \to I_{\text{L}} \left( {r,t} \right)$ at $x \to \infty$. The amount of energy, which is absorbed in a unit volume of a cell during a time step of duration $\Delta t$, is then equal to $\Delta E = \Delta t\alpha I$. This quantity is used to update thermal velocities of heavy particles in the cell according to the equation

$${\mathbf{v}}_{i}^{*} = {\mathbf{u}} + \left( {{\mathbf{v}}_{i} - {\mathbf{u}}} \right)\sqrt {\frac{{E_{\text{a}} + \Delta E}}{{E_{\text{a}} }}} ,$$

(6)

so that the total thermal energy density and temperature of heavy particles in the cell calculated based on the updated velocities $\mathbf{v}_{i}^{*}$ are equal to $E^{*} = E + \Delta E$ and $T^{*} = E^{*} /\left( {\left( {3/2} \right)n_{\text{a}} k_{\text{B}} } \right)$.

The value of $E^{*}$ and number densities of each species $n_{s}$ are then used to find the updated fractions of ions $X_{\left( s \right)}^{z**}$ and temperature $T^{ * *}$ in the cell based on the solution of the Saha equations coupled with the equations that express the energy balance, local neutrality of plasma, and conservation of the heavy particle number for every species. The full set of equations used to predict $X_{\left( s \right)}^{z**}$ and $T^{ * *}$ is presented in Appendix 2. At the final stage of computations, values of $X_{i}^{z}$ and $T_{e,i}$ for all simulated particles in the cell are updated with values of $X_{\left( s \right)}^{z**}$ for corresponding species and $T^{ * *}$, while the velocities of heavy particles are changed according to the equation

$${\mathbf{v}}_{i}^{**} = {\mathbf{u}} + \left( {{\mathbf{v}}_{i}^{*} - {\mathbf{u}}} \right)\sqrt {\frac{{T^{ * *} }}{{T^{ *} }}} .$$

(7)

This approach enforces thermalization of the whole system of particles in each cell by the end of the time step, so that $T_{\text{a}}^{**} = T_{\text{e}}^{**} = T^{ * *}$.

Appendix 2. Models of ionization and absorption of laser radiation

In this work, the simulations are performed based on the equilibrium model of ionization. This model is applied at each time step in every cell of the computational mesh in order to enforce the equilibrium degrees of ionization by the end of the time step. In every cell at each time step of the computational algorithm, we determine, based on parameters of individual simulated particles, the number densities of individual species $n_{s}$ and total thermal energy density $E$ (energy of the chaotic motion of all particles plus the energy of ionization per unit volume). The molar fraction of particles of species $s$ with ion charge $z$, $X_{\left( s \right)}^{z}$, and temperature, $T$, are determined based on the solution of the Saha equations, e.g., [14]

$$\frac{{n_{\text{e}} X_{\left( s \right)}^{z} }}{{X_{\left( s \right)}^{z - 1} }} = \left( {\frac{{2\pi m_{\text{e}} k_{\text{B}} T}}{{h^{2} }}} \right)^{3/2} \exp \left( { - \frac{{{\text{IP}}_{s}^{z} }}{{k_{\text{B}} T}}} \right),\quad s = 1,2, \;z = 1, \ldots ,Z_{\text{max} } ,$$

(8)

where $n_{\text{e}}$ = $\sum\nolimits_{s} {n_{s} } \sum\nolimits_{z} z X_{\left( s \right)}^{z}$, $h$ is the Planck constant, and $m_{\text{e}}$ is mass of an electron. Equation (8) is coupled with the energy conservation equation [14]

$$E = \mathop \sum \limits_{s = 1}^{2} n_{s} \left[ {\left( {1 + \mathop \sum \limits_{z = 1}^{{Z_{\text{max} } }} zX_{\left( s \right)}^{z} } \right)k_{\text{B}} T + \mathop \sum \limits_{z = 1}^{{Z_{\text{max} } }} IP_{s}^{z} X_{\left( s \right)}^{z} } \right].$$

(9)

Equations (8) and (9) at fixed $E$ and $n_{s}$ are solved iteratively with respect to $T$ and $X_{\left( s \right)}^{z}$ using the Newton–Raphson method.

The values of $T$ and $X_{\left( s \right)}^{z}$ are then used to calculate the plasma absorption coefficient $\alpha$ based on the model suggested in Ref. [33] in the form

$$\alpha = \mathop \sum \limits_{s = 1}^{2} \left( {\alpha_{{{\text{PI}}\left( s \right)}} + \alpha_{{{\text{IB}},i\left( s \right)}} + \alpha_{{{\text{IB}},n\left( s \right)}} } \right),$$

(10)

where $\alpha_{{{\text{PI}}\left( s \right)}}$, $\alpha_{{{\text{IB}},i\left( s \right)}}$, and $\alpha_{{{\text{IB}},n\left( s \right)}}$ accounts for the effects of photoionization and inverse Bremsstrahlung in electron collisions with ions and neutral atoms of species $s$:

$$\alpha_{{{\text{PI}}\left( s \right)}} = \mathop \sum \limits_{z = 0}^{{Z_{\text{max} } }} \mathop \sum \limits_{{j = N_{*\left( s \right)}^{z} }}^{{N_{\text{max} \left( s \right)}^{z} }} n_{j\left( s \right)}^{z} \sigma_{{{\text{PI}},j\left( s \right)}}^{z} \left( T \right),$$

(11)

$$\alpha_{{{\text{IB}},i\left( s \right)}} = \left[ {1 - \exp \left( { - \frac{hc}{{\lambda k_{\text{B}} T}}} \right)} \right]\sqrt {\frac{2\pi }{{3m_{\text{e}} k_{\text{B}} T}}} \frac{{4e^{6} k_{\text{e}}^{3} \lambda^{3} n_{\text{e}} n_{s} }}{{3hc^{4} m_{\text{e}} }}\mathop \sum \limits_{z = 1}^{{Z_{\text{max} } }} z^{2} X_{\left( s \right)}^{z} ,$$

(12)

$$\alpha_{{{\text{IB}},n\left( s \right)}} = \left[ {1 - \exp \left( { - \frac{hc}{{\lambda k_{\text{B}} T}}} \right)} \right]\sigma_{{{\text{IB}},n\left( s \right)}} \left( T \right)n_{\text{e}} n_{s}^{0} .$$

(13)

Here $k_{\text{e}}$ is the Coulomb constant, $c$ is the speed of light, $j$ is the energy level, $N_{{ *\left( {\text{s}} \right)}}^{z} = N_{{ *\left( {\text{s}} \right)}}^{z} \left( \lambda \right)$ is the minimum energy level number of an ion of species $s$ and charge $z$ which can be further ionized to charge $z + 1$ by absorbing a photon of wavelength $\lambda$, $N_{\text{max} \left( s \right)}^{z}$ is the maximum energy level considered in calculations, and $n_{j\left( s \right)}^{z}$ is the number density of particles in energy level $j$, which is calculated based on the Boltzmann equilibrium distribution, $n_{j\left( s \right)}^{z} = n_{{0\left( {\text{s}} \right)}}^{z} \left( {g_{j\left( s \right)}^{z} /g_{{0\left( {\text{s}} \right)}}^{z} } \right)\exp \left[ { - \left( {E_{j\left( s \right)}^{z} - E_{{0\left( {\text{s}} \right)}}^{z} } \right)/\left( {k_{\text{B}} T} \right)} \right]$, where $g_{j\left( s \right)}^{z}$ and $E_{j\left( s \right)}^{z}$ are the statistical weight and energy at level $j$.

The photoionization cross section $\sigma_{{{\text{PI}},j}}^{z} \left( T \right)$ is calculated as [33]

$$\sigma_{{{\text{PI}},j\left( s \right)}}^{z} \left( T \right) = \frac{{32\pi^{2} \left( {z + 1} \right)^{2} e^{6} k_{\text{e}}^{3} }}{{3\sqrt 3 h^{4} c\nu^{3} g_{j\left( s \right)}^{Z} }}u_{\left( s \right)}^{z + 1} \left( T \right)\frac{{{\text{d}}E_{j\left( s \right)}^{z} }}{{{\text{d}}j}},$$

(14)

where $\nu = c/\lambda$ is the laser radiation frequency,

$$u_{\left( s \right)}^{z} \left( T \right) = \mathop \sum \limits_{j = 0}^{{N_{\text{max} \left( s \right)}^{z} }} g_{j\left( s \right)}^{z} \exp \left( - {\frac{{ {E_{j\left( s \right)}^{z} - E_{0\left( s \right)}^{z} } }}{{k_{\text{B}} T}}} \right).$$

(15)

is the partition function, and ${\text{d}}E_{j\left( s \right)}^{Z} /{\text{d}}j$ is the spacing between energy levels. The electron-neutral collision cross section $\sigma_{{{\text{IB}},n\left( s \right)}} \left( T \right)$ in Eq. (13) is calculated as suggested in Ref. [34]:

$$\sigma_{{{\text{IB}},n\left( s \right)}} \left( T \right) = \mathop \smallint \limits_{0}^{\infty } K_{a\left( s \right)} \left( E \right)f_{\text{MB}} \left( {E,T} \right){\text{d}}E,$$

(16)

where

$$K_{{{\text{a}}\left( s \right)}} \left( E \right) = \frac{{2e^{2} k_{\text{e}} }}{{3\pi m_{\text{e}} c\nu^{2} }}\sqrt {\left( {\frac{{2\left( {E + h\nu } \right)}}{{m_{\text{e}} }}} \right)} \frac{E + h\nu }{h\nu }\sigma_{{{\text{T}}\left( s \right)}} \left( {E + h\nu } \right),$$

(17)

$f_{\text{MB}} \left( {E,T} \right) = 2\sqrt {E/\left( {\pi \left( {k_{\text{B}} T} \right)^{3} } \right)} \exp \left[ { - E/\left( {k_{\text{B}} T} \right)} \right]$ is the Maxwell–Boltzmann distribution, and $\sigma_{{{\text{T}}\left( s \right)}} \left( E \right)$ is the momentum transport cross section for electron-neutral collisions.

Based on preliminary simulations, the value of $Z_{\text{max} } = 3$ is adopted for all simulations, since the molar fraction of triply ionized atoms does not exceed 1% under conditions considered in this work. The ionization energies are equal to ${\text{IP}}_{1}^{1}$ = 7.73 eV, ${\text{IP}}_{1}^{2}$ = 20.29 eV, and ${\text{IP}}_{1}^{3}$ = 36.84 eV for copper [35] and ${\text{IP}}_{2}^{1}$ = 15.76 eV, ${\text{IP}}_{2}^{2}$ = 27.63 eV, and ${\text{IP}}_{2}^{3}$ = 40.74 eV for argon [36]. The tables with energy levels and their statistical weights for copper and argon ions are adopted from Refs. [35, 36]. The values of the momentum transport cross section $\sigma_{{{\text{T}}\left( s \right)}} \left( E \right)$ for copper and argon atoms in the tabulated form are taken from Refs. [37, 38], respectively, and then interpolated for calculations of $\sigma_{{{\text{IB}},n\left( s \right)}} \left( T \right)$ according to Eqs. (16) and (17).