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\AtBeginDocument{{\noindent\small
Ninth MSU-UAB Conference on Differential Equations and Computational
Simulations.
\emph{Electronic Journal of Differential Equations},
Conference 20 (2013), pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
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\begin{document} \setcounter{page}{1}
\title[\hfilneg EJDE-2013/Conf/20/ \hfil Hydrodynamic modeling]
{Hydrodynamic modeling of $\mathrm{ns}$-laser ablation}
\author[D. Autrique, V. Alexiades, H. Khanal \hfil EJDE-2013/conf/20 \hfilneg]
{David Autrique, Vasilios Alexiades, Harihar Khanal}
\address{David Autrique \newline
Physics, Optimas Research Center - TU Kaiserslautern,
67653 Kaiserslautern, Germany}
\email{dautriqu@physik.uni-kl.de}
\address{Vasilios Alexiades \newline
Mathematics, University of Tennessee,
Knoxville, TN 37996, USA}
\email{alexiades@utk.edu}
\address{Harihar Khanal \newline
Mathematics, Embry-Riddle Aeronautical University
Daytona Beach, FL 32114, USA}
\email{Harihar.Khanal@erau.edu}
\thanks{Published October 31, 2013.}
\subjclass[2000]{92C45, 35K60, 65M99}
\keywords{Nanosecond laser ablation; hydrodynamic model; multiphase; plasma}
\begin{abstract}
Laser ablation is a versatile and widespread technique, applied in an
increasing number of medical, industrial and analytical applications.
A hydrodynamic multiphase model describing nanosecond-laser ablation
(ns-LA) is outlined. The model accounts for target heating and mass
removal mechanisms as well as plume expansion and plasma formation.
A copper target is placed in an ambient environment consisting of helium
and irradiated by a nanosecond-laser pulse. The effect of variable laser
settings on the ablation process is explored in 1-D numerical simulations.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}\label{S:1}
Due to its high spatial and temporal coherence, the laser is an excellent tool
for precise material processing.
The locally deposited laser energy can be used to remove material from
the irradiated sample. This technique, known as laser ablation, is
nowadays used in an increasing number of medical, industrial as well as
analytical applications such as surgery, ophtamology,
nanoparticle manufacturing, pulsed laser deposition and micromachining,
among many others \cite{Bauerle2011, Chrisey1994, Miller1998, Montaser1998}.
In certain industrial applications such as pulsed laser deposition
\cite{Chrisey1994} or analytical applications such as ``nanosecond laser
ablation inductively coupled plasma mass spectrometry''
(ns-LA-ICP-MS), the ablation process takes place inside a chamber
filled with an inert gas \cite{Montaser1998}. In case of LA-ICP-MS for
instance, this ambient gas will transport the ablated matter through a
tubing system towards the analysis equipment \cite{Montaser1998}.
This experimental situation will be studied in the present work,
with copper (Cu) being the target material and helium (He) the ambient gas.
It should be stressed that laser ablation results in a cascade of tightly
coupled processes which depend on the laser parameters, sample characteristics
as well as on the ambient environment.
In a first step the target material will heat up, melt, evaporate and start
to expand into the ambient environment.
Subsequently the laser will trigger breakdown in the expanding vapor;
the plume will partially ionize and a hot plasma is created above
the sample surface. Since this plasma can absorb a significant amount of laser
light, it will shield the target from the incoming laser beam and therefore
reduce the amount of laser energy deposited in the sample. Such back-coupling
effects render the experimental identification of the mechanisms underlying
laser ablation problematic. Therefore modeling and simulation are useful,
as they allow a separate study of the intricately coupled processes.
\emph{Nanosecond} laser ablation has been investigated by means of
kinetic \cite{Garrelie1999study, Itina1997,Kools1993monte};
hydrodynamic \cite{Aden1993, Aghaei2008, Alexiades2010enthalpy,
Anisimov2002_selected, Autrique2012multiphase, Balazs1991, Bulgakov1995,
Clair2011, Ho1995, Le2000, Mazhukin2007_mod,
Qaisar2003, Singh1990, Wen2007a};
as well as hybrid models such as \cite{Harilal2003, Itina2002}.
Because kinetic and hybrid models require long calculation times, one tries
to employ hydrodynamic modeling if at all possible.
Note that a macroscopic description of laser-material removal should treat
the response of the material to variable pressure, energy and density.
Therefore a multiphase approach is required
\cite{Anisimov2002_selected, Qaisar2003,autrique2013role} that accounts for the physical
mechanisms in as well as above the irradiated target.
Hence a hydrocode should account for target heating and the various ablation
mechanisms as well as plume expansion and plasma formation.
In this paper we outline such a multiphase hydrodynamic model and present
results of hydrodynamic simulations exploring the effect of variable laser
wavelength and irradiance.
In \S \ref{S:2} we briefly describe some issues that arise during the
modeling of nanosecond laser ablation.
In \S \ref{S:3} we outline some aspects of the hydrodynamic model that
addresses these issues.
In \S \ref{S:4}, 1-D hydrodynamic simulations for ns-laser ablation of a
copper target (Cu) in a background gas (He), are presented.
Calculations were done for a 6 ns laser pulse, operating in the UV-VIS
regime at wavelengths of 266 nm and 532 nm, respectively and
at laser fluences up to 10 J/cm$^2$. Final conclusions are given in \S \ref{S:5}.
\section{Nanosecond laser ablation}\label{S:2}
During and after the laser-material interaction, a complex cascade of tightly
connected processes occurs in the vicinity of the target surface.
As the copper sample heats up during ns-laser irradiation, a thin layer of
material melts and finally vaporizes.
At near-critical temperatures, the surface region of the material can arrive
in a metastable liquid state. Here homogeneous nucleation
results in the formation of bubbles in the hot melt: the material will
thermally fragment and decompose into a mixture consisting of vapor and liquid.
As soon as the target material arrives in a (super)critical state, it experiences
a liquid-plasma transition \cite{Bulgakova2001_phas1,Likalter1996}.
The laser will trigger breakdown in the ablated material and a plasma starts
to form. This plasma will absorb a part of the incoming laser light, and
therefore shield the target from the laser beam. In the present work, laser
induced absorption due to single and multi-photon ionization
and inverse Bremsstrahlung are
considered \cite{Radziemski1989, Zeldovich2002}
(see also section \S \ref{S:3:2}).
Due to these absorption mechanisms, a hot and quickly expanding plume is formed.
The plasma pressure will increase and exceed the surface pressure of the material.
Under such conditions the phase transition at the surface reverses and the
material condenses back on the sample surface. Moreover the plasma pressure
acting on the molten material can result in melt expulsion: molten material
moves out of the irradiated material volume \cite{Allmen1995, Voisey2003}.
At later times, typically several microseconds after the laser pulse, small
particles are formed during the cooling stage of the plasma. Here homogeneous
nucleation and recondensation processes in the plume result in the generation
of nanosized droplets \cite{Gnedovets1999model, Tillack2004effect}.
In order to describe such a variety of processes, an effective hydrodynamic
model should possess the following features:
\begin{itemize}
\item describe laser heating and account for the various ablation mechanisms,
\item account for variable thermophysical and optical properties,
\item treat plume expansion in an ambient environment,
\item deal with plasma formation under non- equilibrium conditions,
\item capture strong shocks via high resolution numerical schemes,
\item be applicable in 1D, 2D as well as 3D.
\end{itemize}
Additional challenges in laser ablation modeling arise from:
\begin{itemize}
\item extreme space and time scales,
\item extreme variation in material properties,
\item the need for extensive datasets,
\item the need for a wide-range multiphase equation of state.
\end{itemize}
In the next section, we outline some aspects of such a hydrodynamic model.
For simplicity, and consistency with the 1-D numerical results,
we present a 1-D version of the corresponding equations.
\section{Multiphase hydrodynamic model}\label{S:3}
\subsection{Target}
\label{S:3:1}
In modeling ns-laser ablation, one can assume that the laser energy
absorbed by the electrons is instantaneously redistributed and passed to the
lattice. The electron and lattice subsystems of the target achieved thermal
equilibrium; therefore target heating can be described by an
ordinary heat conduction equation. Because material is removed (ablated)
from the target, its surface will move with a certain recession velocity
$v_{\rm rec}$. Hence, it is computationally convenient
to apply a Lagrangian formulation; target heating is described in a reference
frame attached to the receding surface:
\begin{equation}
\frac{{\partial U}}{{\partial t}} - {v_{\rm rec}}(t)
\frac{{\partial U}}{{\partial x}}
= \frac{\partial}{{\partial x}} k(T) \frac{{\partial T}}{{\partial x}}
+ {S_{\rm las}}(x,t)\,.
\label{eq:3:U}
\end{equation}
In \eqref{eq:3:U}, $U$ denotes the internal energy density, $v_{\rm rec}$
is the surface recession velocity, and $k(T)$ represents the
temperature dependent thermal conductivity \cite{Ho1972}.
The laser source term $S_{\rm las}$, at position $x$ and time $t$, is given by
\begin{equation}
S_{\rm las} (x, t) = I(t) (1-R) \alpha e^{- \alpha x} \,,
\label{eq:3:Slas}
\end{equation}
Here $R(T)$ and $\alpha(T)$ denote the reflection and absorption coefficients
of the target, respectively. They are calculated by fitting dielectric
functions that account for intraband as well as interband transitions
\cite{Ashcroft1971interband} to tabulated data \cite{Palik1997}.
The laser intensity (irradiance)
at the target surface, $I_{\rm a} (t)$
is assumed to follow a Gaussian
temporal profile with with standard deviation $\sigma$:
\begin{equation}
I_{\rm a} (t) = \beta I_{\rm max} e^{ - (t-t_{\rm max})^2 / 2\sigma^2 }\,.
\label{eq:3:intens}
\end{equation}
In \eqref{eq:3:intens}, $I_{\rm max}$ denotes the maximum intensity
(input parameter, see \S \ref{S:4}), whereas $\beta$ represents the
total shielding coefficient at the target surface that depends on the
laser wavelength, plume temperature as well as on the species densities
(neutrals, ions and electrons) above the target.
At all times the laser-heated material tends to adjust its pressure
$P_m$ to the ambient pressure $P_{amb}$.
Pressure relaxation can be expressed as follows:
\begin{equation}
\frac{{\partial P_m }}{{\partial t}} -
{v_{\rm rec}}(t)\frac{{\partial P_m }}{{\partial x}}
= - \frac{ { P_m - P_{amb}}}{\tau_{\rm mech} (x,t) } \,.
\label{eq:3:Pm}
\end{equation}
In \eqref{eq:3:Pm}, $\tau_{mech}$ expresses the mechanical relaxation time;
i.e., the time needed for a pressure wave to reach the target surface.
Hence, one obtains $\tau_{mech}(x,t) = x / c_{\rm snd}(x,t)$, where $x$
and $c_{\rm snd}$ are the spatial coordinate and the local speed of sound
in the target. For metals, the minimum mechanical relaxation time is
typically in the order of 10 ps \cite{Chimier2008_effect}.
Since the target material will expand or compress locally after pressure
relaxation, the density changes. Hence, the corresponding material
velocity $v(x,t)$ can be obtained from the continuity equation for given $\rho$:
\begin{equation}
\frac{{\partial \rho }}{{\partial t}} -
{v_{\rm rec}}(t)\frac{{\partial \rho }}{{\partial x}}
= - \frac{{\partial \rho v}}{{\partial x}} \,.
\label{eq:3:rho}
\end{equation}
Mass and energy conservation, Eq.\eqref{eq:3:U}, \ref{eq:3:rho},
are valid mathematically, in weak sense, irrespective of the material
phase \cite{Alexiades1992mathematical}.
At each timestep, the internal energy density $U$,
material pressure $P_m$, mass density $\rho$, and material velocity
$v$ are updated to new time. In a next step, the corresponding phase state
and material temperature are found.
The system of equations Eq.\eqref{eq:3:U}-\ref{eq:3:rho} is closed by a
multiphase equation of state (EOS) \cite{Levashov2007_eos}.
A projection of the equation of state on the $\rho - T$ plane is shown
in Figure \ref{fig:phdiag}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{ Phase diagram of copper projected on the $\rho - T$ plane.
In addition to the stable material phases (solid, liquid, gas and plasma),
the metastable liquid and 2-phase (liquid+gas) regions, are represented.
The respective phase boundaries (binodal, spinodal, melting and sublimation
curves) are indicated. In addition, two isobars are shown that correspond
to the mininimum (0 GPa) and maximum pressures (1 GPa) encountered during
the simulations}
\label{fig:phdiag}
\end{figure}
Note that the recession velocity $v_{\rm rec}$ is one of the unknowns in the
problem. In our model, three ablation mechanisms and corresponding recession
velocities are distinguished, which are found as part of the numerical solution:
(1) surface ablation due to evaporation and condensation, $v_{\rm ev}$;
(2) volumetric mass removal due to bubble formation in the metastable state,
$v_{\rm bub}$;
and (3) volumetric ablation in the (super)critical region, $v_{\rm crit}$ .
%
The recession velocity $v_{\rm rec}$ in \eqref{eq:3:U} can therefore be
formulated as:
%
\begin{equation}
v_{\rm rec}(t)=v_{ev}(t)+v_{\rm bub}(t)+v_{\rm crit}(t) \,.
\label{eq:3:recvel}
\end{equation}
Hence \eqref{eq:3:recvel} will depend on the material state and accordingly
on the material temperature, pressure and density.
\subsection{Plume and Plasma}
\label{S:3:2}
The ablation mechanisms result in the creation of a dense plume that
expands above the copper surface in the ambient gas, here helium.
The expanding plume is modeled by a set of Euler equations:
\begin{gather}
\frac{\partial \rho}{\partial t} = - \frac{\partial \rho v}{\partial x}
\label{rho}\\
\frac{\partial \rho _v}{\partial t} = - \frac{\partial \rho _v v}{\partial x}
\label{rho_v} \\
\frac{\partial \rho v}{{\partial t}} = - \frac{\partial \rho {v^2} + P }{{\partial x}}
\label{rhov} \\
\frac{{\partial \rho \varepsilon }}{{\partial t}}
= - \frac{\partial \left( \rho \varepsilon+ P \right) v}{{\partial x}}
- \frac{\partial q}{{\partial x}} + S_{\rm{IB}} + S_{\rm{PI}}-S_{\rm{rad}}
\label{rhoenergy}
\end{gather}
These equations express mass \eqref{rho}, momentum \eqref{rhov} and energy
conservation \eqref{rhoenergy} in the plume domain.
Here $\rho=\rho_{v} + \rho_{b}$ denotes the total mass density,
with ${\rho _v}$ and ${\rho _b} $ the partial densities of vapor and
background gas, respectively. The total momentum density and the total energy
density of plasma are $\rho v$ and $\rho \varepsilon$ respectively.
$P$ denotes the total pressure, whereas $q$ is the heat flux transported
by the various species in the plasma.
In calculating $q$, an expression for the thermal conductivity is applied,
which accounts for degeneracy as well as ionization effects
\cite{Lee1984conductivity}.
The resulting numerical heat fluxes are limited using physical heat flux
limiters, i.e. particles cannot move faster than their thermal velocity.
Note that species diffusion in the plume is not considered here due to the
disparate time scales of the phenomena; diffusion processes become
important much later, at near-$\mu s$ times \cite{Le2000}.
The laser will trigger breakdown in the dense expanding vapor formed above
the target surface. Laser induced breakdown is modeled by a collisional
radiative model, presented in the accompanying article by
Khanal et al \cite{Khanal2013}. The collisional radiative model accounts
for the various collisional and radiative processes in the plasma, such as one-,
two- and three-photon ionization, radiative decay, electron impact
excitation and ionization, as well as the respective recombination
reactions \cite{Chung2005flychk,Mazhukin2003optical,Morel2010modeling}.
It describes the initial non-equilibrium state encountered in the plasma
during the laser pulse. A set of rate equations (described in \cite{Khanal2013})
expresses the temporal evolution of the electron density and temperature,
as well as the atomic level populations in the expanding plume.
The equations account for the various collisional and radiative processes
in the plasma. Since the species densities will drop quickly during the
expansion process, it is reasonable to describe the plasma in the ideal
gas approximation. Hence, the closure relations for the plasma are provided
by the ideal gas equation of state.
The collisional radiative model provides the three source terms encountered
in the energy equation (Eq.\ref{rhoenergy}). Here $S_{\rm IB}$ and
${S_{\rm PI}}$
are the energy source terms due to inverse Bremsstrahlung (IB) and
one-, two-, and three-photon ionization (PI), respectively, whereas
${S_{\rm rad}}$ is the radiation loss of the plasma \cite{Zeldovich2002}.
After the breakdown stage, the collisional radiative model indicates
that the (spatially averaged) plasma attains about equal electron,
excitation, and heavy species temperatures, which means it approaches
a state close to Local Thermodynamic Equilibrium (LTE).
When the plume reaches a state close to LTE, the rate equations are
switched off. Under LTE, the atomic levels become Boltzmann distributed and
a local state is achieved where electrons, ions as well as the excited states
are characterized by the same temperature $T$.
At that instant the plasma is descibed in the ideal gas approximation and
the temperature, electron and ion densitities are retrieved from the
Saha-Eggert equations \cite{Zeldovich2002}:
\begin{equation}
\frac{{{n_e}{x_i}}}{{{x_{i - 1}}}}
= \frac{{{u_i}(T)}}{{{u_{i - 1}}(T)}}{\Big( {\frac{{2\pi {m_e}kT}}{{{h^2}}}}
\Big)^{3/2}}\exp \Big( { - \frac{{{I_i}}}{{kT}}} \Big)
\label{saha}
\end{equation}
Here ${n_e}$ denotes the total electron number density, ${x_i}$ ${x_{i - 1}}$
are the molar fractions of ions at states $i$ and $i-1$, respectively, $m_e$
is mass of electron, $k$ is the Boltzmann constant, $h$ is the Planck constant.
${I_i}$ is the ionization potential of the ion with respect to state $i-1$,
${u_i}(T)$ is the electronic partition function for an ion in state
$i$ \cite{Ralchenko2008nist}.
The Saha equations and corresponding closure relations are solved iteratively
by a Newton-Raphson method to calculate the unknowns ${n_{e}}$,
${x_i}$ and $T$.
\subsection{Computational Strategy}
\label{S:3:3}
The target is discretized into a non-uniform grid, with high nodal density
near the surface.
The smallest target cells (of size 6 nm) are placed in the near-surface region.
They resolve the absorption length $1/ \alpha(T)$ over which laser energy is
deposited (see \eqref{eq:3:Slas}).
Initially, the plume region is discretized into a uniform grid of
length $\Delta x$ = 500 nm. Adaptive mesh refinement is used to
resolve the strong gradients encountered in the plume domain
\cite{Tang2003adaptive}. The mesh in the plume domain is refined till
the grid reaches a lower limit of 50 nm.
Moreover, the grid expands ahead of the heat wave in the
target domain and ahead of the expanding plasma in the plume domain.
A central second order
advection scheme is applied \cite{Kurganov2000new} , while the temporal
evolution of the problem proceeds through explicit time-stepping
(see the accompanying article \cite{Khanal2013}).
Variable time-stepping is employed respecting the CFL condition in the
entire computational domain.
Time steps are of order $10^{-13}$ s. The total simulation time
is 50 ns, well beyond the duration of a single laser pulse.
\section{Numerical Simulations}
\label{S:4}
We present 1-D simulations of laser ablation of copper (Cu) in a background
gas (He) under variable laser settings. The copper target and the background
gas reside initially in a stationary state and
are initialized at standard pressure and temperature (STP). The entire
system (target+background gas) is irradiated by a ns-laser pulse of
6 ns FWHM (Full Width at Half Maximum).
The laser operates in the UV-VIS regime at wavelengths of 266 nm or 532 nm
and at variable irradiances
( $4 \times 10^{12}$ - $1.6 \times 10^{13}$ $W/m^2$ ).
The employed irradiances, correspond to fluences that vary between
2 and 10 J/cm$^2$.
In the following sections, numerical simulations are shown that represent
the evolution of the target and plasma plume for the laser settings mentioned
above.
In \S \ref{S:4:1}-\ref{S:4:3}, results are presented for laser pulses
operating at variable wavelengths and fixed maximum peak irradiances
of $10^{13}$ $W/m^2$, whereas in \S \ref{S:4:4} a full parametric
study is employed for the calculation of plasma transmissivities and
ablation depths.
\subsection{Target Heating}
\label{S:4:1}
The temporal evolution of the surface temperature and laser intensities
(irradiances) at the target surface are depicted in Fig.\ref{fig:surfint}(a).
Two laser pulses with a peak irradiance of $10^{13}$ $W/m^2$ (black curve)
interact with the copper surface. The first laser pulse operates in the
ultraviolet region, at 266 nm, whereas the second produces visible,
monochromatic light at 532 nm. During the laser-material interaction the
copper target will heat up very fast. Around 7 ns the surface temperature
exceeds the normal boiling point ($T_{boil} =$ 2836 K) and evaporation becomes
significant. At that instant, a plume consisting of evaporated copper develops
above the target surface. After 8 ns, the material reaches near-critical
temperatures ($T_{crit}=7800 K$). Now the material arrives in a metastable
state near the binodal (not shown here) and will fragment thermally.
A lot of material enters the plume domain in a short time and starts to expand
above the surface.
The laser triggers breakdown of the material in the dense plume and a plasma
is formed above the surface.
The plasma will shield the target from the laser and reduce the laser intensity
at the surface.
The actual intensities at the target surface are shown for 266 nm (blue curve )
and 532 nm (red curve), respectively, which clearly exhibits a decreasing trend
of temperature and intensity at the surface after 8 ns.
However, since the laser light still reaches the target surface, the heating
process proceeds; around 9 ns the copper target arrives in the supercritical
state. The target experiences a liquid-plasma transition and material is ejected
in the plume domain.
The mass density near the target surface increases, whereas the plasma responds
through laser absorption and subsequent expansion; as a result
oscillations are observed in the laser irradiance patterns as seen in
Figure \ref{fig:surfint}(a).
Figure \ref{fig:surfint}(b) depicts time dependent profiles of the reflectivities
and the surface temperature for both cases. Since the material will expand
during the laser heating, its electron density decreases and the reflectivity
starts to drop. Around 10 ns, the material arrives in a supercritical state;
the strong reduction of the electron density results in a drastic change
of the optical properties; the hot melt near the surface becomes quasi-transparent. At the end of the laser pulse, the situation is reversed: the surface temperature decreases and the reflectivity increases again.
Although the initial reflectivity of the copper target, irradiated by a
266 nm pulse is half the value of the reflectivity in the 532 case, similar
minima are encountered around 10 ns.
Hence, the amount of laser energy reaching the target is similar in both
cases (see \eqref{eq:3:Slas}) and the target properties behave alike for
both laser wavelengths.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig2a} \quad
\includegraphics[width=0.48\textwidth]{fig2b} \\
(a) \hfil (b)
\end{center}
\caption{ Dashed curves represent surface temperature for
$\lambda$ = 266 nm(blue) and for $\lambda$ = 532 nm (red).
The peak irradiance is the same in both cases ($I_{\rm max}=10^{13}$ $W/m^2$).
Solid curves depict:
(a) laser intensity:
original intensity (black) and actual intensities for $\lambda$ = 266 nm (blue) and for
$\lambda$ = 532 nm (red).
(b) reflectivity: for $\lambda$ = 266 nm (blue) and for
$\lambda$ = 532 nm (red)}
\label{fig:surfint}
\end{figure}
\subsection{Plasma Formation}
\label{S:4:2}
The onset of laser induced breakdown is modelled by a collisional radiative
model (see \S \ref{S:3:2}); computational details can be found in the
article of Khanal et al \cite{Khanal2013}.
During the laser action, photon ionization (PI) triggers breakdown of
the gaseous copper in the vapor above the target surface.
As a result, a significant amount of ions and electrons is formed in the
expanding plume. The electrons will speed up in the laser field and
collide with the present neutrals and ions; at this stage the laser-plasma
absorption proceeds through Inverse Bremsstrahlung (IB).
Figures \ref{fig:alpha}(a-b) represent temporal profiles for spatial
averaged absorption coefficients at wavelengths of 266 and 532 nm, respectively.
The laser pulse duration and intensity are the same in both cases
($FWHM=6 ns$, $I_{\rm max}=10^{13}$ $W/m^2$). Although the total absorption
coefficient ($\alpha_{TOTAL}$), attains similar values in both cases, the
absorption coefficients of the underlying processes ($\alpha_{\rm PI}$
and $\alpha_{\rm IB}$), are clearly different.
In case of 266 nm laser irradiation (see Fig.\ref{fig:alpha}(a) ),
$\alpha_{MPI}$ dominates during the entire time.
In the 532 nm case, however, Inverse Bremsstrahlung due to electron-ion
collisions ($\alpha_{\rm IB}$), dominates after 9.5 ns
(see brown solid line in Fig.\ref{fig:alpha}(b)). This effect can be
attributed to the fact that the efficiency of (multi) photon ionization
absorption increases at decreasing wavelengths ($\sim 1/\lambda$), whereas
the absorption coefficients for electron-ion Inverse Bremsstrahlung scale
as the cube of the laser wavelength ($\sim \lambda^3$).
Since the total absorption coefficients are similar, the total amount of
energy invested in the plasma will be about the same. Therefore, the overall
behavior of the plasma will be similar as well.
This observation indicates that one has to be careful with the interpretation
of experimental data in such a situation; although similar plasma behavior
would be observed here, the underlying absorption mechanisms are clearly
different.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig3a} \quad
\includegraphics[width=0.48\textwidth]{fig3b} \\
(a) \hfil (b)
\end{center}
\caption{Time evolution of spatially averaged absorption coefficients
in the plasma is shown for the same peak irradiance
($I_{\rm max}=10^{13}$ $W/m^2$).
Solid lines represent absorption due to photo-ionization
($\alpha_{MPI}$), electron-neutral and electron-ion inverse
Bremsstrahlung ($\alpha_{\rm IB}$) as well as the total absorption
coefficient ($\alpha_{TOTAL}$):
(a) for $\lambda$ = 266 nm
(b) for $\lambda$ = 532 nm
} \label{fig:alpha}
\end{figure}
\subsection{Plume Expansion}
\label{S:4:3}
During the ablation process a lot of ablated copper enters the plume domain
and a high density region near the target surface is formed
(Figure \ref{fig:density}(a)).
The dense plume will expand above the surface, while it pushes the
ambient helium gas away (see Figure \ref{fig:density}(b)).
Both the ablated material and the background gas pile up near their interface.
This effect is known as the ``snowplow effect'' \cite{Itina2002}.
Laser induced breakdown results in a hot plasma
(Figure \ref{fig:temperature}(a)) that expands at speeds in the order
of 10 km/s above the target surface (Figure \ref{fig:temperature}(b)).
Since the plume pressure above the target surface increases, a situation
is encountered where the plume pressure exceeds the pressure at the target
surface.
Hence material moves back towards the target and condenses on the surface.
This results in negative velocities in the vicinity of the target
(Figure \ref{fig:temperature}(b)).
After the end of the laser pulse, here around 16 ns, the temperature
and velocity will decrease as the plasma expands
(see Figure \ref{fig:temperature}(a-b)).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig4a} \quad
\includegraphics[width=0.48\textwidth]{fig4b} \\
(a) \hfil (b)
\end{center}
\caption{
Spatial profiles of species number densities at different times
for the same laser settings as in Fig.\ref{fig:alpha}.
Since the results for 266 and 532 nm are very similar,
only the case $\lambda=$ 266 nm is shown.
(a) number density of ablated material (copper);
(b) number density of background gas (helium)
} \label{fig:density}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig5a} \quad
\includegraphics[width=0.48\textwidth]{fig5b}\\
(a) \hfil (b)
\end{center}
\caption{
Spatial temperature (a) and velocity (b) profiles
at various times, for 266 nm laser pulse operating
at peak irradiance $10^{13}$ $W/m^2$
} \label{fig:temperature}
\end{figure}
\subsection{Ablation depths and Transmission data }
\label{S:4:4}
Figures \ref{fig:crater}(a) and \ref{fig:crater}(b) show
calculated ablated depths and transmissivities at two wavelengths
and for fluences (i.e. integrated irradiances) up to 10 J/cm${}^2$.
In the 266 nm case, higher ablated depths are found, when compared to the
532 nm case (see Fig.\ref{fig:crater}(a)). Since the 266 nm laser will
penetrate deeper into the target material, more material will be removed
than in the 532 nm case (see \ref{fig:crater}(a)).
The plasma formed above the surface will shield the laser from the target.
Since the surface integrals of the total absorption coefficients are about
the same for both wavelengths, more plasma shielding will occur
in the situation where more ``absorbers'' (species) are encountered, here
the 266 nm case.
Thus, the transmissivities exhibit the opposite behavior, the 266 nm
wavelength generates more plasma shielding than the 532 nm one.
Note that transmissivities represent the amount of laser energy that was
distributed between plasma and target.
In case the transmissivities would deviate from transmission experiments
such as \cite{Vorob'ev1985, Bulgakov1999thermal, Clair2011},
it would mean that the model fails to predict correct energy balances in
the domain and automatically implies that the underlying physics needs to be
revised.
Since mass removal is governed by a complex cascade of processes that occur
in and above the target, comparison of calculated and measured ablation depths
are an additional valuable step in the validation of the model.
Experimental verification studies are planned in the near future.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig6a}\quad
\includegraphics[width=0.48\textwidth]{fig6b}
(a) \hfil (b)
\end{center}
\caption{
Transmission data and ablated depths for variable laser
settings. Simulations were performed for a range of
irradiances corresponding to fluences up to 10 J/cm$^2$.
Results are for wavelengths 266 nm (blue) and 532 nm (red):
(a) Ablated depths
(b) Transmissivity
} \label{fig:crater}
\end{figure}
\section{Conclusion}
\label{S:5}
%
A hydrodynamic multiphase model for laser ablation of copper in an ambient
environment consisting of helium is presented.
During laser irradiation, several complex, tightly coupled physical processes
occur in and above the target.
Surface as well as volumetric ablation mechanisms, result in the formation of
a dense vapor plume above the surface.
The laser triggers breakdown in the vapor and a hot, quickly expanding plasma
is created above the target.
The onset of laser induced breakdown in the vapor is modelled, employing a
collisional radiative model.
As soon as a state close to Local Thermodynamic Equilibrium is achieved,
the collisional radiative model is switched off and the plasma description
proceeds through the solution of Saha-Eggert equations.
Numerical results show the evolution of the target and plasma plume at various
laser settings.
The laser operates in the UV-VIS regime at wavelengths of 266 nm or 532 nm,
and at fluences up to 10 J/cm$^2$.
Despite the fact that both wavelengths result in similar behavior of the target
and plasma, it is found that the underlying absorption mechanisms in the plasma
are different.
Whereas photo-ionization dominates the entire absorption process in the case
of 266 nm laser irradiation, Inverse Bremsstrahlung processes compete with
photo-ionization under 532 nm irradiation.
Although similar plasma behavior could be experimentally observed in such
a case, the underlying mechanisms can clearly be different.
Our results indicate that the interpretation of experimental data in such a
situation might be much more complicated than it would seem at first sight.
Hence, both modeling and experimental verification are required for a detailed
understanding of the mechanisms underlying ns-laser ablation.
\subsection*{Acknowledgements}
The authors wish to thank P. Levashov, K. Khishenko and M. Povarnitsyn
for the equation-of-state data set and advice.
The first author (DA) acknowledges financial support from the Deutsche
Forschungsgemeinschaft (Emmy Noether-Program, grant RE 1141/11).
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\end{document}