Pulsed laser evaporation: equation-of-state effects

Appl. Phys. A 69, 617–620 (1999) / Digital Object Identifier (DOI) 10.1007/s003399900131

Applied Physics A Materials Science & Processing  Springer-Verlag 1999

Pulsed laser evaporation: equation-of-state effects S.I. Anisimov1 , N.A. Inogamov1 , A.M. Oparin1 , B. Rethfeld2 , T. Yabe3 , M. Ogawa3 , V.E. Fortov4 1 Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia 2 Institute for Theoretical Physics, Technical University Braunschweig, Braunschweig, D-38106, Germany 3 Department of Energy Sciences, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 226, Japan 4 High Energy Density Research Center, Joint Institute for High Temperatures, Russian Academy of Sciences, 127412

Moscow, Russia

Received: 2 March 1999/Accepted: 28 May 1999/Published online: 21 October 1999

Abstract. Theoretical study of laser ablation is usually based on the assumption that the vapor is an ideal gas. Its flow is described by gas dynamics equations [1, 2]. The boundary conditions at vaporization front are derived from the solution of the Boltzmann equation that describes the vapor flow in the immediate vicinity of the vaporizing surface (so-called Knudsen layer) [1]. This model is applicable within the range of temperatures much lower than the critical temperature of target material. In the present work, a general case is considered when the temperature of the condensed phase is comparable to or higher than the critical temperature. The dynamics of both condensed and gaseous phases can be described in this case by the equations of hydrodynamics. The dynamics of vaporization of a metal heated by an ultrashort laser pulse is studied both analytically and numerically. The analysis reveals that the flow consists of two domains: thin liquid shell moving with constant velocity, and thick low-density layer of material in two-phase state. PACS: 81.60Z; 65.70.+y; 64.60.Ht At moderate laser intensities, laser-produced evaporation is adequately described by a simple model [1, 2] assuming that (i) a sharp boundary separates the condensed and gaseous phase; (ii) the atoms emitted from the phase boundary have nonequilibrium (“semi-Maxwellian”) velocity distribution; (iii) the equilibrium is attained within the Knudsen layer whose thickness is of the order of the mean free path of atoms. Light absorption in the vapor plume can usually be neglected. This model is applicable when the temperature of the condensed phase is much lower than the critical temperature of the target material. The vapor density in this case is low, therefore the vapor flow within the Knudsen layer can be described by the Boltzmann kinetic equation. Solving this equation yields the boundary conditions for vapor flow in the plume [1]. At higher laser intensities and relatively long pulses, light absorption in the plume plays an important role. The temperature of the target in this case can be considerably higher than the critical temperature, and a sharp boundary between

the condensed and gaseous phase disappears. The vapor density becomes comparable to the density of condensed material, and its flow can be described by the equations of gas dynamics [3] supplemented with a wide-range equation of state [4]. Short-pulse laser vaporization of condensed matter has a number of specific features. In a metal, visible and infrared laser radiation is absorbed mainly by conduction electrons. Energy transfer from electrons to the crystal lattice proceeds rather slowly due to the great difference between electron and ion mass (or, what is the same [5], due to the difference between the sound speed and electron Fermi velocity). According to [5, 6], the characteristic time of lattice heating for different metals falls in the range 1 to 100 ps. This means that, during a ps laser pulse, the absorbed energy is stored in the electron subsystem and the major mechanism of energy transfer is the electron heat conduction. Hydrodynamic motion at this stage can be neglected. It is easy to see that hydrodynamics comes into play at later times when the electrons come into equilibrium with √ In √the lattice. fact, the electron thermal wave travels as z T ∼ χt ∼ νe tτtr , where χ ∼ νe2 τtr , τtr is the electron momentum relaxation time (the so-called transport time), and νe is the √mean electron velocity ( νe equals the Fermi velocity, νF = 2F /m, if the electrons are degenerate, and the thermal velocity in nondegenerate case). The propagation of hydrodynamic perturbations follows the law z H ∼ ct , where c is the sound speed. Thus the hydrodynamic motion becomes the major mechanism of energy transfer when t > τtr νe2 /c2 ∼ τtr M/m ∼ τei , where M and m are the ion and electron mass, respectively and τei is the electron–ion energy relaxation time. We can use, therefore, a one-temperature hydrodynamic model when considering the evaporation of a metal absorbing ultrashort laser pulse. Since for ps and sub-ps pulses the evaporation takes place after the end of the laser pulse, the target is not shielded by the vapor, and the initial temperature of the condensed phase may be much higher than in the case of a ns pulse of comparable fluence. Specifically, the temperature well above the critical point can be reached in the surface layer. The thickness of heated surface layer by the time the hydrodynamic motion comes into play is of the order of max(cτei , µ−1 ), where µ is the absorption coefficient of

618 a metal. This layer expands then into vacuum or ambient gas. In the present paper we will consider the structure of the expansion flow in the case of two-phase equation of state (EOS). We will show that in contrast to the ideal gas expansion with constant adiabatic exponent, in the two-phase system a complicated density profile is formed with sharp density gradient. Note that two-phase effects play an important role in laser experiments because, for typical ablation pressure of the order of a few megabars the expansion adiabatic enters the twophase region in the phase diagram. 1 Self-similar rarefaction wave. Numerical solution To simplify the problem under consideration, let us consider one-dimensional isentropic expansion into vacuum of a uniform semi-infinite layer of metal whose density equals the normal density, and whose temperature is of the order of the critical temperature of the metal. The flow is described by the equations of gas dynamics ∂%/∂t + ∂(%u)/∂z = 0 , ∂u/∂t + u∂u/∂z(1/%)∂ p/∂z = 0 ,

(1)

where standard notations are used. The set of equations (1) should be supplemented by the equation of isentrope S( p, %) = const which we take in the form p = p(%). Equations (1) can be transformed [7] to new variables, P and M:

Fig. 2. Density as a function of mass velocity in self-similar rarefaction wave

condensed material u = I = 0, we set P(ξ) = 0. Therefore, u(ξ) = −I(ξ). Equation (3) yields ξ = u − c = −I(%) − c(%) .

(4)

If the z axis is directed toward the vacuum, the function P(ξ) can be shown to be a constant. Since in the unperturbed hot

The latter equation determines the functions %(ξ) and u(ξ) in the implicit form. The results of the numerical calculations performed in this way are shown in Figs. 1 and 2. Figure 1 represents the density profile as a function of similarity variable ξ. Figure 2 shows the dependence of the density on the mass velocity u. The calculations were carried out for aluminum at normal initial density %0 = 2.7 g/cm3 and several initial temperatures (the values of the initial temperature and entropy are given in upper right corner). The equation of state of aluminum was constructed using the method described in [4, 8]. As is seen from Figs. 1, 2, a characteristic feature of the flow is a sharp front appearing on the vacuum side of expanding material. The origin of density jump can be readily understood if we look at Fig. 3 where the expansion isentropes in the ( p, V) plane are presented (V = 1/%). When an isentrope enters the two-phase region, the slope of the isentrope (which is proportional to the square of the sound speed) abruptly decreases by several orders of magnitude. Thus, within the two-phase region the similarity vari-

Fig. 1. Density profile in self-similar expansion wave

Fig. 3. Expansion isentropes of aluminum in ( p, V) plane

∂P/∂t + (u + c)∂P/∂z = 0 , ∂M/∂t + (u − c)∂M/∂z = 0 ,

(2)

R%0 where P = u + I, M = u − I, I = c(%)d%/%, and c(%) = % p (∂ p/∂%)S is the sound speed. In the case under study the flow is self-similar [7]. The similarity variable is ξ = z/t. Equations (2) take the form: (−ξ + u + c)P 0(ξ) = 0 , (−ξ + u − c)M 0 (ξ) = 0 .

(3)

619 able ξ = −I(%) − c(%) ≈ −I(%) is essentially independent of the density. 2 Self-similar rarefaction wave. Analytical model To qualitatively describe the expansion flow we substitute a real isentrope (similar to those shown in Fig. 3) by two polytropes pi = Ci %γi (i = 1, 2), where the subscript i = 1 refers to condensed phase and i = 2 to the two-phase system. The two polytropes should be matched at the point e, where the isentrope crosses the curve of phase equilibrium (dashed curve in 1/2 Fig. 3). The sound speed c = (∂ p/∂%)S has a discontinuity at the point e. It is clear that c1e  c2e . Consider a self-similar flow. Equations (1) reduce in this case to cu 0 + N(u − ξ)c0 = 0 , (u − ξ)u 0 + Ncc0 = 0 ,

(5)

0

where u = du/dξ and N = 2/(γ − 1). A non-trivial solution of (5) exists if the determinant of the system equals zero, i.e., if (u − ξ)2 = c2 , or ξ = u ± c. In our case ξ = u − c. Substituting this equation into (5), we obtain the general solution [7]: u = Nξ/(N + 1) + A , c = −ξ/(N + 1) + A ,

(6)

much less than unity, whereas in standard situations the exponent γ = (∂ log p/∂ log %)S > 1. To calculate the constant A in (6) we make use of the sound speed c2e and similarity variable ξ2e on right side of the point e. The calculation yields: u 2 (ξ) = [N2 (ξ + c2e ) + N1 (c0 − c1e )]/(N2 + 1) ≈ 2ξ − N1 (c0 − c1e ) , c2 (ξ) = [−ξ + N2 c2e + N1 (c0 − c1e )]/(N2 + 1) ≈ ξ − N1 (c0 − c1e ) .

(9)

Here we took into account that c2e ≈ 0 and N2 ≈ −2. Using the equation for isentrope and (9) we obtain the density profile in the second portion of the rarefaction wave:   −ξ + u e + N2 c2e N2 %2 (ξ) = %e × ≈ c2e (N2 + 1) %e [1 + (ξ − ξ2e )/c2e ]−2 ,

(10)

where %e is the density at the plateau. Because c2e is very small, the density sharply decreases in the vicinity of ξ2e , as it is seen in Figs. 1, 2. At large z the density is proportional to z −2 . 3 Target of finite thickness. Reflection of rarefaction wave

where N1 = 2/(γ1 − 1) and c0 is the sound speed in the undisturbed hot material. Equations (7) describe the first (left) portion of the rarefaction wave located in the interval −c0 < ξ < ξ1e . In Fig. 3 this portion corresponds to the left branch of the isentrope connecting the initial point a and the point e of intersection of an isentrope with the curve of phase equilibrium (binodale). At the point e, the first portion of the rarefaction wave should be matched with the second portion that is described by the polytrope with the exponent γ2 . According to (4), at the matching point

Up to this point we considered the expansion into vacuum of a semi-infinite heated material layer. The above solutions are valid until the rarefaction wave advances into a uniform unperturbed material. In this section we consider the expansion of a uniformly heated target of finite thickness 2h. In such a target two rarefaction waves run toward the center of the target and reflect from the center at time instant tr = h/c0 . Note that a similar flow structure arises when a uniformly heated film of thickness h is located on a cold substrate. The flow in the vicinity of the center is determined by the interaction of the incident and reflected rarefaction waves. It is convenient to consider a qualitative picture of wave interaction assuming that the material is an ideal gas with adiabatic exponent γ = 3. For this value of γ the set of equations for Riemann invariants is reduced to two independent equations which can be readily solved [9]. This simple model qualitatively describes the dynamics of condensed material. In the region of the reflected wave the solution reads:

ξ1e = u 1e − c1e ,

c = h/t ,

where A is an arbitrary constant. To determine this constant we take into account that the rarefaction wave is adjacent to the undisturbed hot solid at rest. This yields u 1 (ξ) = N1 (ξ + c0 )/(N1 + 1) , c1 (ξ) = (−ξ + N1 c0 )/(N1 + 1) ,

ξ2e = u 2e − c2e .

(7)

(8)

Since the velocity u is continuous at the matching point e, (8) yield: ξ2e > ξ1e . We see thus that the first and second portions of the rarefaction wave are not immediately adjacent to one another. They are separated by a plateau — a uniform domain moving with the velocity u e = u 1e = u 2e . Employing (7) one can readily show that u e = N1 (c0 − c1e ) . The width of the plateau is ξ2e − ξ1e = c1e − c2e ≈ c1e . The plateau is clearly seen in Figs. 1 and 2. Let us consider now the second portion of the rarefaction wave. Note that for this portion the adiabatic exponent γ2 is

% = %0 h/c0 t = %0 tr /t ,

u = z/t .

(11)

According to (11), the density in the region of wave interaction does not depend on the coordinate z and decreases with t increasing. The mass velocity is a linear function of z. This flow is called the uniform deformation flow. We see that the interaction of incident and reflected rarefaction waves results in the formation of a “hole” in the central part of the expanding cloud. Thus, the expanding cloud at t > tr includes three elements: (i) the “hole” in the central part; (ii) the plateau with constant density; (iii) the low-density tail on the vacuum side of the plateau. This qualitative picture is confirmed by numerical solution of the gas dynamics equations. The calculation was performed for an aluminum target. The equation of state of Al was taken from the SESAME library [10]. At the

620 ents observed in [11] are attributable to very low sound speed in the two-phase region. The Newton rings observed a few ns after the absorption of an ultrashort laser pulse may be explained as the result of the interference of the probe beams reflected from the sharp external boundary of the plateau and from the relatively cold shock-compressed internal part of the target. Acknowledgements. The authors are grateful to Prof. J. Meyer-ter-Vehn, Prof. D. von der Linde, and Dr. K. Sokolowski-Tinten for helpful discussions. One of the authors (S.A.) is grateful to the Laboratory of Nuclear Reactors, Tokyo Institute of Technology for offering the guest position. This work was supported by the Russian Fund for Fundamental Research, grants 98-2-17441 and 97-2-16044.

References

Fig. 4. Density profile resulting from the interaction of incident and reflected rarefaction waves. Numerical calculation

initial moment the density of the target was %0 = 2.7 g/cm3 , temperature T0 = 6000 K, sound speed c0 = 6.6 km/s, and thickness of the target 2h = 80 nm. Figure 4 shows the density profile at t = 50 ps. The characteristic features of the flow are clearly seen in Fig. 4. The density in the plateau region is about one half of the initial solid-state density. In the region of the central “hole” the material is in the two-phase state. The spikes within the “hole” region result from rough smoothing of tabulated EOS data. The picture described above is in qualitative agreement with recent experimental data [11]. The sharp density gradi-

1. S.I. Anisimov: Sov. Phys.- JETP 27, 182 (1968) 2. S.I. Anisimov, Y.A. Imas, G.S. Romanov, YuV. Khodyko: Action of High Power Radiation on Metals (National Tech. Inform. Service, Springfield, VA 1971) 3. A.V. Bushman, I.V. Lomonosov, V.E. Fortov: Equations of State of Metals at High Energy Densities (Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka 1992) (in Russian) 4. S.I. Anisimov, V.A. Gal’burt, M.F. Ivanov, I.E. Poyurovskaya, V.I. Fisher: Sov. Phys.- Tech. Phys. 24, 295 (1979) 5. M.I. Kaganov, I.M. Lifshits, L.V. Tanatarov: Sov. Phys.- JETP 4, 173 (1957) 6. P.B. Allen: Phys. Rev. Lett. 59, 1460 (1987) 7. L.D. Landau, E.M. Lifshits: Fluid Mechanics (Pergamon Press, Oxford 1982) 8. V.A. Agureikin, S.I. Anisimov, A.V. Bushman, G.I. Kanel’, V.P. Karyagin, A.B. Konstantinov, V.F. Minin, S.V. Razorenov, R.Z. Sagdeev, S.G. Sugak, V.E. Fortov: High Temp. 22, 761 (1984) 9. K.P. Stanyukovich: Nonstationary Motion of Continuous Media (Nauka, Moscow 1978) (in Russian) 10. B.I. Bennet, J.D. Johnson, G.I. Kerley, G.T. Rood: LANL Rep. LA7130, 1978 11. D. von der Linde, K. Sokolowski-Tinten, J. Bialkowski: Appl. Surf. Sci. 109/110, 1 (1996)