atoms Review Shock Waves in Laser-Induced Plasmas Beatrice Campanella, Stefano Legnaioli, Stefano Pagnotta , Francesco Poggialini and Vincenzo Palleschi * Applied and Laser Spectroscopy Laboratory, ICCOM-CNR, 56124 Pisa, Italy; [email protected] (B.C.); [email protected] (S.L.); [email protected] (S.P.); [email protected] (F.P.) * Correspondence: [email protected]; Tel.: +39-050-315-2224 Received: 7 May 2019; Accepted: 5 June 2019; Published: 7 June 2019 Abstract: The production of a plasma by a pulsed laser beam in solids, liquids or gas is often associated with the generation of a strong shock wave, which can be studied and interpreted in the framework of the theory of strong explosion. In this review, we will briefly present a theoretical interpretation of the physical mechanisms of laser-generated shock waves. After that, we will discuss how the study of the dynamics of the laser-induced shock wave can be used for obtaining useful information about the laser–target interaction (for example, the energy delivered by the laser on the target material) or on the physical properties of the target itself (hardness). Finally, we will focus the discussion on how the laser-induced shock wave can be exploited in analytical applications of Laser-Induced Plasmas as, for example, in Double-Pulse Laser-Induced Breakdown Spectroscopy experiments. Keywords: shock wave; Laser-Induced Plasmas; strong explosion; LIBS; Double-Pulse LIBS 1. Introduction The sudden release of energy delivered by a pulsed laser beam on a material (solid, liquid or gaseous) may produce a small (but strong) explosion, which is associated, together with other effects (ablation of material, plasma formation and excitation), with a violent displacement of the surrounding material and the production of a shock wave. The time evolution of a laser-induced shock wave and its geometry are dominated by the energy of the laser and the shape of the plasma generated. If the energy of the laser is delivered in a small spot, we may have spherical or hemispherical shock waves, while in case of elongated plasma (in liquids, for example) the shock wave may assume a cylindrical shape. Intermediate situations can be more complex (see Figure1). Atoms 2019, 7, 57; doi:10.3390/atoms7020057 www.mdpi.com/journal/atoms AtomsAtoms 20182019, 6, x, 7FOR, 57 PEER REVIEW 2 of 214 of 14 Figure 1. Spherical, hemispherical and cylindrical (with hemispherical caps) shock waves induced by a Figurelaser 1. in Spherical, a gas, solid hemispherical surface or inside and cylindrical a liquid. (with hemispherical caps) shock waves induced by a laser in a gas, solid surface or inside a liquid. The theoretical basis for the study of shock wave generation and propagation were laid by studies ofThe the blasttheoretical wave generatedbasis for inthe nuclear study explosions.of shock wave The studiesgeneration performed and propagation at Los Alamos were by Bethelaid by et al. studiesin 1941 of the [1] (publishedblast wave in generated a 1947 in in a reportnuclear then explosions. declassified The in studies 1958) and performed in UK by at the Los British Alamos physicist by BetheG. I.et Taylor al. in [19412] (published [1] (published in 1950) in evidenceda 1947 in a the report main then characteristics declassified of in a nuclear1958) and explosion in UK versusby the an Britishordinary physicist explosion, G. I. Taylor i.e., the [2] fact (published that the in energy 1950) isevidenced released inthe a main nuclear characteristics explosion in of a verya nuclear limited explosionvolume. versus The theoryan ordinary describing explosion, the blast i.e., the wave fact produced that the energy by a nuclear is released explosion in a nuclear is thus explosion the theory in aof very a point limited strong volume. explosion. The theory In Reference describing [1], thethe theoreticalblast wave treatment produced of by the a nuclear point strong explosion explosion is thuswas the developed theory of bya point John strong von Neumann. explosion. While In Re realizingference [1], the the di ffitheoreticalculties of treatment an exact treatment of the point of the strongshock explosion wave propagation, was developed von Neumann,by John von taking Neumann. into account While therealizing dimensions the difficulties of the explosion of an exact source, treatmentdemonstrated of the shock thata wave great propagation, simplification von of theNeumann, model cantaking be obtainedinto account by considering the dimensions the sourceof the of explosionenergy assource, a point. demonstrated that a great simplification of the model can be obtained by consideringIn this the approximation, source of energy a self-similar as a point. solution to the propagation of the shock waves can be searched for,In assumingthis approximation, a sudden release a self-similar of an energy solution E0 in to a negligiblethe propagation time and of volume. the shock waves can be searchedAssuming for, assuming an adiabatic a sudden expansion release of of an the energy (spherical) E0 in shocka negligible wave, time a purely and dimensionalvolume. reasoning leadsAssuming to the determinationan adiabatic expansion of the expansion of the (spheric law ofal) the shock shock wave, wave a as: purely dimensional reasoning leads to the determination of the expansion law of the shock wave as: !1/5 E R(t) = 0 / t2/5, (1) / () = K(γ)ρ0 , (1) () where R(t) is the radius of the shock wave at time t, ρ0 is the unperturbed density of the gas and K(γ) where R(t) is the radius of the shock wave at time t, is the unperturbed density of the gas and ()is ais constant a constant which which depends depends on the on adiabatic the adiabatic coefficient coefficient g of the g gas. of Similarthe gas. studies Similar were studies developed were in developedthe same in period the same in the period Soviet in Unionthe Soviet by L.I. Union Sedov. by L.I. Sedov. SedovSedov extended extended the theself-similar self-similar treatment treatment to stro to strongng explosions explosions produced produced by bylinear linear (cylindrical (cylindrical waves)waves) or planar or planar (plane (plane waves) waves) sources. sources. An Animmediate immediate generalization generalization of Equation of Equation (1) (1)to these to these shock shock wavewave geometries geometries leads leads to the to the expression: expression: /()!1/(2+α) () = E0 /()2/(2+, α) (2) R(t) =() t , (2) K(γ)ρ0 with α = 3 for spherical shock waves, α = 2 for cylindrical and α = 1 for plane shock waves. with α = 3 for spherical shock waves, α = 2 for cylindrical and α = 1 for plane shock waves. The work of Sedov, originally published in a Russian journal [3], later became available in the The work of Sedov, originally published in a Russian journal [3], later became available in the Western world (in 1959) with the publication of the famous book Similarity and Dimensional Methods Western world (in 1959) with the publication of the famous book Similarity and Dimensional Methods in Mechanics [4]. In fact, the description of the shock wave propagation is often referred to the Sedov– in Mechanics [4]. In fact, the description of the shock wave propagation is often referred to the Taylor point strong explosion theory. Sedov–Taylor point strong explosion theory. An interesting discussion on the limits of the Taylor approach, which obtained some popular success after the successful determination of the (classified) energy of the first atomic bomb tested in Atoms 2019, 7, 57 3 of 14 AtomsAn 2018, interesting 6, x FOR PEER discussionREVIEW on the limits of the Taylor approach, which obtained3 some of 14 popular success after the successful determination of the (classified) energy of the first atomic bomb tested in 19451945 inin thethe Nevada Nevada desertdesert (see(see Figure Figure2 ),2), is is reported reported by by Michael Michael A.B. A.B. Deakin Deakin in in Reference Reference [ 5 [5].]. According toAccording Deakin, Taylor’sto Deakin, calculation Taylor’s calculation of the Trinity of the bomb Trinity energy bomb [ 6energy] (which, [6] (which, in the word in the of word his biographer of his G.K. Batchelor,biographer causedG.K. Batchelor, him to becaused “mildly him to admonished be “mildly admonished by the USArmy by the forUS Army publishing for publishing his deductions his from deductions from their (unclassified) photographs” [7]) was indeed overestimated, although by a mere their (unclassified) photographs” [7]) was indeed overestimated, although by a mere 3% with respect 3% with respect to the calculations that could have been derived by the exact treatment of von toNeumann the calculations or Sedov. that could have been derived by the exact treatment of von Neumann or Sedov. Figure 2. 2. TheThe blast blast wave wave produced produced by bythethe first first nuclear nuclear test,test, 16 ms 16 after ms afterthe explosion. the explosion. The radius The radiusof of the the wave is about 200 m. (Trinity test, New Mexico 1945). wave is about 200 m. (Trinity test, New Mexico 1945). The critical point of the discussion is the evaluation of the K constant in Equation (1), which is a The critical point of the discussion is the evaluation of the K constant in Equation (1), which is function of the adiabatic coefficient γ of the gas in which the shock wave propagates. While the a function of the adiabatic coefficient γ of the gas in which the shock wave propagates. While the treatment of von Neumann and Sedov is exact, the determination of the K constant using Taylor’s treatmentapproach is of approximated. von Neumann One andshould Sedov consider, is exact, in any the case, determination that Equation (1) of always the K constantunderestimates using Taylor’s approachthe total energy is approximated.