Scaling Laws of an Exploding Liquid Column under an Intense Ultrashort X-Ray Pulse

Scaling Laws of an Exploding Liquid Column under an Intense Ultrashort X-Ray Pulse Alfonso M. Gañán-Calvo Departamento de Ingeniería Aerospacial y Mecánica de Fluidos, ETSI, Universidad de Sevilla, Camino de los Descubrimientos s/n 41092, Sevilla, Spain (Received 25 November 2018; published 7 August 2019) A general formulation of the partial destruction of a liquid object in vacuum after the sudden deposition of a very large amount of energy is proposed. That energy instantaneously raises the pressure of a portion of the liquid to extreme values and changes its state, which causes its explosive expansion into vacuum and against the rest of the liquid object. When the deformable object is a liquid capillary column, the model reduces to a universal equation for the evolution of the expanding gap between the two sides of the exploding liquid column. The theoretical analysis contemplates two asymptotic stages for small and large times from the initiation of the blast, whose asymptotic solutions are fitted to available experimental data. A universal approximate analytical solution is obtained. A complete dimensional analysis of the problem and an optimal collapse of experimental data reveal that the proposed solution is in remarkable agreement with experiments of a jet exploding after being irradiated by an ultrashort and intense x-ray pulse from an x-ray free electron laser. DOI: 10.1103/PhysRevLett.123.064501 The interaction of high energies with matter is a subject of fundamental importance in applied physics [1,2]. There is ample research literature on the reaction of condensed matter to the localized deposition of a very large energy density by diverse means (e.g., by large electric shocks, laser and ion beams, etc.), especially in the area of inertial confinement fusion. Extensive related research on the hydrodynamic processes due to the very fast local vaporization, like shock-driven hydrodynamics, liquid compressibility phenomena, bubble implosion, etc., appears in the literature for spherical [3,4] and cylindrical geometries [5,6]. In these latter cases, the energy fluxes ranged from 1010 to 1012 W=cm2. The advent of powerful energy sources like the free electron lasers (XFELs) has raised the energy fluxes above 1022 W=cm2, with the top > 1023 W=cm2mark announced by the project ELI-NP at Magurele [7], the largest known so far in our planet. The local ultrafast (femtoand attosecond-scale) deposition of these extreme energy densities have been crucial to observe and test new phenomena and more diverse geometries [8], overcoming the effects of the energy release history of previous slower deposition means [5,6]. The physics of intense blasts against deformable (liquid) objects is of fundamental importance in areas as varied as nanosurgery [9], serial femtosecond crystallography (SFX) [10], the study of extreme physical properties and strange phases of matter [11,12], or testing the equations of state of matter from cosmology [13,14] to the processing or production of new materials [15]. In general, the extremely large local pressures appearing in the liquid in very short times and the need to quantify them are crucial objectives of research. When the liquid is a closed object with free surfaces (e.g., a sphere or a rod [8]), the scaling laws of those pressures, the input energy fraction going to pressure, the evolution of the blast, or the blast shape factor against the liquid being destructed need to be determined to predict the effects. In SFX, the damage caused to the samples upstream of the blast is the subject of increasing attention [16]. In this work we focus on the destruction of a cylindrical rod of liquid (e.g., a capillary jet) after the rapid local deposition of a very large energy density [8] in processes like the analysis or processing of materials with synchrotron, SFX, etc. This deposition causes an explosion that expands partially in vacuum and partially against the liquid rod in the axial direction, violently dividing the cylinder into two sections. Unlike the radial actions that occur in exploding wires, causing strong radial compression phenomena only [5], the explosions here studied imply a geometrical degree of freedom in the axial direction that is absent in the former. This implies the appearance of a variety of new physical phenomena [8]. The essential difference with other processes where a liquid cylinder breaks up is the extreme value of the power locally supplied: it can be trillions times larger than that of natural mechanical methods (e.g., capillary breakup, or the sudden jet obstruction or its transverse disruption with a projectile). Here we propose a general formulation of the problem, applying it to a cylindrical liquid column. The physics (hydrodynamics) of the blast is described obtaining the expansion velocities, the peak compression stresses undergone by the liquid column, and the energies involved. General compact formulation of blasts in partial contact with deformable objects.—Consider an amount of condensed deformable matter Msurrounded by vacuum PHYSICAL REVIEW LETTERS 123, 064501 (2019) Editors' Suggestion 0031-9007=19=123(6)=064501(5) 064501-1 © 2019 American Physical Society (Fig. 1). In this work, that matter is in the form of a cylindrical rod. An amount of energy EDexceeding by orders of magnitude that to vaporize a portion Mgof that matter is suddenly injected in it. As a consequence, it violently expands as a blast both into vacuum and against M(Fig. 1). The expanding hot gas domain VgðtÞcan be consider to comprise two virtual subdomains VpðtÞand VeðtÞ(from now on pushing and expanding volumes, respectively) separated by a fluid surface SiðtÞ, as formally defined in the Supplemental Material [17]: (i) VpðtÞpushes and deforms Mby dominant pressure forces, (ii) the mass and energy of VeðtÞare constant by definition while it expands into vacuum, and (iii) VeðtÞis assumed charge neutral. By virtue of two latter conditions, VeðtÞdoes not make any work on M. Those definitions do not impose any artificial restriction on the natural evolution of the total gas domain VgðtÞ, but they provide a drastic simplification, as follows. Defining Vo¼Vpðt0ÞþVeðt0Þas the initial energized volume at the initial instant t0, the volume fraction χ¼Vpðt0Þ=Vois a fixed problem parameter. Thus, given that the analysis of Veis irrelevant since its energy is constant and its evolution is decoupled from M, one can write the following compact equation of conservation of energy that governs the coupled evolution of M and Vp: Zt t0ZSpþSi Pv·ndAdt0þPoVpðtÞ1−γðχVoÞγ ðγ−1Þ¼χED;ð1Þ where the first term in the left side is the total work made by Vpon Mthrough the fluid surfaces Siand Sp(the surface of Vpin contact with M; see Fig. 2) since the beginning of the blast (vand nare the velocity and unit normal vectors of the surfaces), and the second is the internal energy left in Vp at a given t. The initial conditions are Vpðt0Þ¼χVo, Pðt0Þ¼Po, and PoVo=ðγ−1Þ¼ED. Since the pressure is initially the same for Vpand Ve,χis also the injected energy fraction contained in Vpat the beginning of the )(tSe )(t Si )(tSs vacuum 0 = PD E g M vapor )(tVp )(tVe M 0=P xt/2 j d FIG. 1. General sketch of the problem. Thin, medium, and thick dotted lines indicate SeðtÞ,SiðtÞ, and SsðtÞ, respectively. SpðtÞis the surface of VpðtÞin contact with the liquid. The instantaneous radial velocity xt=2of the expanding lamella is indicated. Ve(t), expanding gas volume with constant mass and energy Vp(t), pushing gas volume with constant mass 8µs12 µs ~5 ns 15 ns 4µs 20 µm x(t) Si(t)Ss(t) Vp(t)Ve(t)Ve(t) Vp(t) Sp(t) FIG. 2. Evolution of the blast induced by a strong x-ray laser pulse (photon energy 8.2 KeV, 0.75 mJ, duration 30 fs) on a liquid microjet of 20 μm discharged in vacuum [8]. The effective beam diameter is approximately 1μm. Snapshot at 5 ns: initial stage, where the highly compressed quasicylindrical pushing volume VpðtÞexpands against the two liquid fronts while the still nearly cylindrical expanding gas volume VeðtÞdoes it in the radial and axial directions. Approximate illustrating shapes of Vpand Veare depicted. 15 ns: Vpstarts expanding in the radial direction, while Vebegins a doughnut-shaped mixed radial-spherical expansion. 4μs: Vpundergoes a mixed radial-spherical expansion. 8μs: both Vpand Vetend to expand spherically. Final stage (t≳12 μs): Both Vpand Vealready expand nearly spherically. Supporting pictures from Stan et al. [8], Supplemental Material. Positions, shape, and size of control volumes are just illustrative. PHYSICAL REVIEW LETTERS 123, 064501 (2019) 064501-2 blast, or the efficiency of the blast against M. The factor (γ−1) can be considered the Grüneisen coefficient of the initial energized matter (in many cases a warm dense matter state [12]), γbeing its adiabatic coefficient when it expands. Consistently with the assumption made in Ref. [8], if the evolution of the gas in the blast is assumed quasi-isentropic this coefficient stays nearly constant from large to small densities along the blast, as it will be shown. The first term in the left-hand side of Eq. (1) is the total energy received by the object in the process up to time t. This compact formulation in terms of a single geometrical variable is advantageous for relatively simple geometries like the explosion of a microjet produced by flow focusing [22], which gently carries samples (as initially suggested in Ref. [19], see Supplemental Material [17]): In that application, the microjet is shot by a train of extremely short xray pulses in SFX [8,10,23]. Application to the destruction of a liquid cylinder.— Consider a liquid cylinder of diameter djwhere a very large energy density is instantaneously deposited in a slice (see Fig. 2). That energy splits the cylinder (a jet, in SFX) in two symmetrical rods whose separating fronts develop two symmetrical expanding liquid lamellas, whose mechanical energy received from the gas can be expressed as Zt t0ZSpþSi Pðxp;t 0Þvp·npdAdt ¼1 2ρl πd2 j 4Zx x0 x2 t 4dx ¼πd2 jρl 32 Zt t0 x3 tdt; ð2Þ where xis the distance between the two separating liquid fronts, subscript tindicates the time derivative, and ρlis the density of the liquid. x0and t0are the initial values of the gap size and time, respectively. For conservation of momentum, we assume in expression (2) that the liquid is radially ejected into the two liquid lamellas at an instantaneous speed xt=2due to the overpressure in Vp. Besides, the ratio of the liquid thermal layer thickness λ to the jet diameter djcan be estimated as λ=dj∼½K2dj= ðρlc2EDÞ1=2, where Kand care the thermal conductivity and specific heat of the liquid. For jet sizes below 100 μm and energies EDused in SFX experiments, λwould be smaller than the molecular size, which would make the thermal energy transfer to the liquid negligible once the rapid blast takes place. This supports neglecting the internal energy gain by the liquid due to diffusion in Eq. (2). Defining the variables ϕ¼x=lo,τ¼t=to, and Ω¼ Vp=ðχVoÞmade dimensionless with the characteristic length lo, time toand volume χVo¼l3 oin Eqs. (1) and (2), Eq. (1) can be expressed in nondimensional form as Zτ τo ϕ3 τdτþχΩðτÞ1−γ¼χ;ð3Þ with its time derivative form as ϕ3 τ¼χðγ−1ÞΩ−γΩτ⇒ϕ2 τ¼χðγ−1ÞΩ−γdΩ dϕ;ð4Þ where we have defined to¼½πρld2 jl3 o=ð32EDÞ1=2.loand χ will be determined from dimensional arguments and maximum correlation of experimental data. On the other hand, although ΩðτÞis an unknown variable of the problem, the mathematical structure of Eq. (4) and physical principles will univocally fix the asymptotic trends of Ω. Geometrically, ϕis related to the shape factor of the object (the column) to the effective blast volume, represented by Ω. In the following, their relationships for both large and small times τare analyzed. Asymptotic behavior of the pushing gas volume Ωfor small and large times τ.—In the very initial stages, both the expanding and pushing volumes Veand Vpproduce an enormous push against the two liquid fronts forming the gap (Fig. 2, time t¼5ns). The geometry of both Veand Vp remains nearly cylindrical for a while, especially that of Vp. Hence, the nondimensional form of the pushing volume should scale as Ω∼ϕsince its expansion would proceed predominantly in the axial direction. This occurs because (i) by definition, the expanding volume cannot be radially pushed by Ωat a higher rate than the opening gap, and (ii) the expanding volume pushes against Ss(the periphery of the radially expanding liquid layer, see Fig. 2)withpressures necessarily smaller than those at Si, preventing a radial expansion of the pushing volume Ω. Therefore, assuming that ϕ∼τα0, using Eq. (4) one should have τ2ðα0−1Þ∼τ−γα0⇒α0¼2 2þγ:ð5Þ On the other hand, in the last stages of the blast (just before surface tension force overcomes the gas pressure, see Fig. 2, t≳12 μs), one should expect that both the expanding and pushing volumes would expand predominantly in the radial direction, which allows a self-similar solution like the one early analyzed by Wedemeyer [21].Thissolutionyieldsthe following radial distribution of the total energy inside the gas sphere (including both Veand Vp): P γ−1þρg v2 2¼PoðBt−1Þ3γ γ−1ð1−ξ2Þ½γ=ðγ−1Þ þγ 2B2B t3 ð1−ξ2Þ½1=ðγ−1Þξ2;ð6Þ where B¼½31=2ðγ−1Þ=2,ξ¼½ρoB=ðγPoÞðr=tÞ,ρois the initial density of the expanding gas, vthe gas speed, and r the radial spherical coordinate from the center of the gap (Fig. 2. See Supplemental Material [17] for additional details on this solution). The fundamental conclusions from this solution are: (i) For ξ≪1(i.e., inside Vpor Ω), the kinetic PHYSICAL REVIEW LETTERS 123, 064501 (2019) 064501-3 energy to pressure ratio becomes as small as ξ2;i.e.,pressure dominates over inertia in Vpas anticipated. (ii) According to the self-similar nature of the solution, the position of the expanding edge of Vpwould correspond to a constant (small) value of ξaccording to mass conservation, for any value ξ≤1 [21], since the gas should move with a constant speed v¼aoξ=B at that edge, while ξ¼1is the expanding edge of Ve. Hence, one should expect Ω∼τ3for τ≫1. Again, assuming that ϕ∼τα1and using Eq. (4) one should have τ3ðα1−1Þ∼τ−3γþ2⇒α1¼5 3−γ:ð7Þ In this limit, onewould have Ω∼ϕ3=ð5=3−γÞ. Interestingly,that solution would demand a logarithmic evolution of the gap for perfect monoatomic gases (γ¼5=3). This is the only scenario for which a logarithmic evolution is contemplated, in contrast to the general logarithmic trend proposed by Stan et al. [8]. In summary, one may approximately express the evolution of the gap xðtÞ¼loϕðτÞas ϕ¼ϕoτα0½1þðτ=τ1Þδðα1−α0Þ=δ;ð8Þ where constants ϕo,τ1, and δshould be obtained by fitting to either experiments or numerical simulations. Finally, lo was expected to be proportional to djin Ref. [8]. However, having defined Vpðt0Þ¼χVo¼l3 oand expecting lo∼x0, from the definition of the initial energized volume Vo¼ x0ðπ=4Þd2 jone can conveniently define lo¼djχ1=2;ð9Þ where the earlier introduced efficiency χshould be a function of the geometry ratio η¼rB=djof the beam radius rBto the jet diameter, and the ratio of initial energy density ED=Vo(see Supplemental Material [17] for SFX) to the energy density of cohesion ρlHv(≃2.3GPa for water), that can be written in terms of the parameter: Πv¼πr2 BdjFðηÞρlHv ED ;ð10Þ where Hvis the heat of vaporization at the temperature of the liquid. FðηÞis a function of the order unity reflecting the shape of the beam and the influence of the beam to jet ratio (see Supplemental Material [17] for details). Usually, one has Πv≪1and therefore a simpler functional dependency as χ¼χðηÞis expected. To verify our model, we have used the experimental results published in Ref. [8]: fourteen combinations of pulse energies and cylinder (jet) diameters, keeping the liquid (water) and beam focus rBconstant. Figure 3(a) depicts the measurements of the gap distance x as a function of time. We use the properties of water at ambient temperature (ρl¼1000 kg=m3,σ¼0.072 N=m, μl¼0.001 Pa s−1) for Stan’s experiments. In their analysis, they readily used djas the reference length, and two possible characteristic times, namely, τlfor a low pressure model, and τhfor a high pressure model [see Supplemental Material [17], Figs. 2(a) and 2(b)]. In our proposal, defining χ¼η2βin Eq. (9), one can investigate the role of the beamto-jet diameter ratio in the characteristic distance lo. We have performed a statistical correlation analysis between τand ϕdata, before the liquid surface tension takes over. We have found that the chi-squared logarithmic differences with local averages is minimized (indicating optimum data collapse: see Supplemental Material [17] for details and quantitative values for the goodness of fit) for β¼0.14 0.003 and Eo¼48.50.5μJ, where Eois the minimum energy at the onset of electrostatic trapping (Supplemental Material [17]). This last value would yield approximately 60 MJ=kg for a beam radius of 0.5μm and 31 MJ=kg for 0.7μm, in agreement with the estimation by Stan et al. (30 MJ=kg). Using those previous best correlation values of βand Eo, in what follows we seek the best fit of the approximate function (8) to the experimental data. The minimum 2.74 µm, 0.75 mJ 3.5 µm, 0.07 mJ 3.5 µm, 0.17 mJ 3.5 µm, 0.35 mJ 3.5 µm, 0.75 mJ 4.28 µm, 0.75 mJ 5.31 µm, 0.75 mJ 5.51 µm, 0.75 mJ 5.6 µm, 0.07 mJ 20 µm, 0.05 mJ 20 µm, 0.07 mJ 20 µm, 0.17 mJ 20 µm, 0.35 mJ 20 µm, 0.75 mJ t(s) (a) (b) x/lo t/to small times asymptotics (dominant liquid reaction) large times asymptotics (dominant gas expansion) x(m) FIG. 3. (a) Dimensional data from Stan et al. [8]. Color codes identify the different jet diameters and pulse energies. (b) Optimal data collapse using to¼½πρld2 jl3 o=ð32EDÞ1=2and lo¼djηβto make times and distances nondimensional. The approximate analytic solution (black line) shows a remarkable fitting to data after optimum collapse. Dot-dashed lines: τ¼t=to≪1(black, small times asymptotics); τ≫1(blue, large times asymptotics). PHYSICAL REVIEW LETTERS 123, 064501 (2019) 064501-4 chi-squared logarithmic difference is obtained for ϕo¼ 0.523 0.002,τ1¼65 0.5, and δ¼1.20.05, with γ¼1.50.01. This supports the hypothesis in Ref. [8] that the Grüneisen coefficient for water is a nearly constant value Γ¼γ−1≃0.5along the process. The approximate function (8) is also plotted in Fig. 3(b) for reference, showing a remarkable fitting. Given the large numerical value of τ1in the solution (8), the analysis of large or small values of τshould be obviously understood as compared to τ1. Moreover, assuming that the evolution starts when the material in Vois already in dense vapor phase and the initial velocity of the liquid fronts is approximately equal to H1=2 v, from Eq. (8) the initial gap would result ϕinit:¼GðηÞΠ−1=γ v, where GðηÞ¼½ðγþ2Þ−2ϕγþ2 oπη2−βFðηÞ1=γ. Finally, the efficiency of the explosions resulted as χ¼ðrB=djÞ2β¼ ðrB=djÞ0.28, where rB=djrange from 0.025 to 0.2 (for dj from 20 to 2.74 μm). In other cases where rB>d j[23], further analysis would provide an extended knowledge of the efficiency χand the additional verification of the physical insights here proposed. Indeed, using rB>d j in SFX would significantly increase the sample hit rate by covering the whole jet section, making measurements insensitive to small accidental drifts of the jet. Claudiu Stan kindly provided the original data sets, and made very valuable suggestions. This work was supported by the Ministerio de Economía y Competitividad (Spain), Plan Estatal Retos, Project No. DPI2016-78887-C3-1-R. Discussions with Pablo Villanueva, Janos Hajdu, Henry Chapman, Anton Barty, Jos´e M. López-Herrera, Francisco Cruz-Mazo, and Roberto Piriz are acknowledged. Suggestions by Pascual Riesco-Chueca are appreciated. [1] S. Agostinelli et al.,Nucl. Instrum. Methods Phys. Res., Sect. A 506, 250 (2003). [2] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012). [3] M. Roth, T. E. Cowan, M. H. Key, S. P. Hatchett, C. Brown, W. Fountain, J. Johnson, D. M. Pennington, R. A. Snavely, S. C. Wilks, K. Yasuike, H. Ruhl, F. Pegoraro, S. V. Bulanov, E. M. 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