The equation of state of solid nickel aluminide
Damian C. Swift, Dennis L. Paisley, Kenneth J. McClellan, Graeme J. Ackland
aa r X i v : . [ c ond - m a t . o t h e r] J un The equation of state of solid nickel aluminide
Damian C. Swift ∗ and Dennis L. Paisley P-24, Physics Division, Los Alamos National Laboratory,MS E526, Los Alamos, New Mexico 87545, U.S.A.
Kenneth J. McClellan
MST-8, Materials Science and Technology Division, Los Alamos National Laboratory,MS G770, Los Alamos, New Mexico 87545, U.S.A.
Graeme J. Ackland
Department of Physics, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland, U.K. (Dated: 2 August 2005, revised 2 November 2005, 16 February 2006, and 30 June 2007 – LA-UR-05-6096)The pressure-volume-temperature equation of state of the intermetallic compound NiAl was cal-culated theoretically, and compared with experimental measurements. Electron ground states werecalculated for NiAl in the CsCl structure, using ab initio pseudopotentials and density functionaltheory (DFT), and were used to predict the cold compression curve and the density of phonon states.It was desirable to interpolate and smooth the cold compression states; the Rose form of compressioncurve was found to reproduce the ab initio calculations well in compression but exhibited significantdeviations in expansion. A thermodynamically-complete equation of state was constructed for NiAl,which overpredicted the mass density at standard temperature and pressure (STP) by 4%, fairlytypical for predictions based on DFT A minimally-adjusted equation of state was constructed bytilting the cold compression energy-volume relation by ∼ µ m thick, to speeds between 100 and600 m/s. Point and line-imaging laser Doppler velocimetry was used to measure the acceleration ofthe flyer and the surface velocity history of the target. The velocity histories were used to deducethe stress state, and hence states on the principal Hugoniot and the flow stress. Flyers and targetswere recovered from most experiments. The effect of elasticity and plastic flow in the sample andwindow was assessed. The ambient isotherm reproduced static compression data very well, and thepredicted Hugoniot was consistent with shock compression data. PACS numbers: 62.20.-x, 62.50.+p, 64.30.+t
I. INTRODUCTION
The equation of state (EOS) relating the pressure,compression, and temperature of solids is important tounderstand the structure of rocky planets and the re-sponse of materials for dynamic loading during impactsor energetic events such as explosions. The EOS alsoprovides a test of our knowledge of underlying physics,particularly the states of electrons under the influenceof the ions, described by many-body quantum mechan-ics. Accurate and thermodynamically-complete EOS arevaluable as it is extremely difficult to measure the tem-perature of material in a shock wave experiment, andtemperature is a key parameter determining the rate atwhich plastic flow, phase transitions, and chemical reac-tions occur.Several approaches have been devised to calculate theEOS essentially from first principles. One way of classi-fying the approaches is into those in which the electronsare treated explicitly, and those in which the effect ofthe electrons is subsumed into effective interatomic po-tentials. Although interatomic potentials can be derivedfrom calculations in which the electrons are includedexplicitly , it can be difficult to represent the non-local aspects of the electron wavefunctions faithfully via aninteratomic potential, and additional complications arisein deriving cross-potentials between dissimilar types ofatom, which are necessary for calculations of compoundsand alloys. With the electrons treated explicitly, someEOS calculations have been made for elements of lowatomic number in which exchange and correlation in theelectron wavefunctions have been treated rigorously us-ing quantum Monte-Carlo techniques , but these calcu-lations are computationally intensive and have spannedrelatively narrow ranges in state space. Almost uni-versally, exchange and correlation between the electronstates have been treated using variants of the Kohn-Shamdensity functional theory (DFT) , which allows theelectrons to be represented efficiently as single-particlestates. Similarly, excitations of the electrons and ionshave been represented with varying rigor. The electronicheat capacity may be treated with simple approxima-tions such as the Sommerfeld model , by populating thedensity of electron energy levels calculated at zero tem-perature, with a density of levels which is calculated con-sistently with the excitations , or it may be ignored al-together in many situations sampling states from roomtemperature to an electron-volt or so. Ionic motion hasbeen incorporated through a Gr¨uneisen model (usuallyfitted to the zero-temperature energy-volume relation) ,by performing simulations of the classical motion of theatoms under the action of interatomic potentials or forcesfrom the electron states , and by calculating the nor-mal modes of the crystal lattice and populating them asphonons.We have previously predicted EOS and phase dia-grams for elements using electron ground states cal-culated using DFT, electron excitations into the zero-temperature band structure, and phonon modes calcu-lated from electronic restoring forces as atoms are dis-placed from equilibrium . An attraction of the methodis that compounds and alloys can in principle be treatedin exactly the same way as elements. We have subse-quently demonstrated that the electron ground states canbe found, and EOS estimated using a Gr¨uneisen treat-ment of the thermal excitations, for the stoichiometricalloy NiTi . A careful study of the ab initio phononmodes has been made at zero pressure for the stoichio-metric alloy NiAl . Here, we report a more rigorous cal-culation of the EOS for NiAl using ab initio phonons and electronic excitations, and comparing with static andshock compression data. We also investigate the use ofan empirical relation for the zero-temperature isothermof NiAl.Shock wave data have been obtained mainly from theimpact of flyers launched by the expansion of compressedgases. In our experiments, the flyer was accelerated bythe expansion of a confined plasma, heated by a laserpulse. The interpretation of shock experiments oftendepends on the properties of other materials in the as-sembly, such as transparent windows, and these may beaffected by time-dependent phenomena such as plasticflow. We discuss the validity of our laser-flyer experi-ments for measuring the EOS of NiAl. II. THEORETICAL EQUATION OF STATE
A thermodynamically complete EOS can be expressedas any thermodynamic potential with respect to its twonatural variables, such as specific internal energy e interms of specific volume v and specific entropy s . Theatomic properties of matter lead naturally to expressionsfor contributions to the internal energy in terms of thevolume (or mass density ρ = 1 /v ) and temperature T .Following our previous study of the EOS of elements , e was notionally split into the cold compression curve e c , lattice-thermal contribution e l , and electron-thermalcontribution e e : e ( ρ, T ) = e c ( ρ ) + e l ( ρ, T ) + e e ( ρ, T ) . (1)The relation e ( ρ, T ) was then used to construct the ther-modynamically complete relation e ( s, v ) and hence otherdesired quantities such as pressure, by invoking the sec-ond law of thermodynamics as described later. The coldcompression energy e c was calculated from the ground state of the electrons with respect to stationary atoms,the lattice-thermal energy e l from the phonon modes ofthe crystal lattice, and the electron-thermal energy fromthe zero-temperature electron band structure though inpractice it was a small contribution to the EOS in thesolid regime. A. Electron ground state calculations
Over the density range of interest, the core electronsare affected relatively little by each atom’s environ-ment, so pseudopotentials were used in preference to all-electron calculations. The electronic ground states werecalculated quantum mechanically using the CASTEPcomputer code. This code implements the plane wavepseudopotential method to solve the Kohn-Sham equa-tions of DFT with respect to the Schr¨odinger Hamil-tonian, to calculate energies and Hellmann-Feynmanforces and stresses.Pseudopotentials were generated using the Troullier-Martins method with 1s2s2p (Al) and 1s2s2p3s3p (Ni)electron shells treated as core. Consequently 13 valenceelectrons per unit cell are considered. NiAl was treatedas non-magnetic.For a cubic material, a sequence of constant volumecalculations suffices to determine the cold compressioncurve. The volume was held fixed while restoring forceswere calculated to determine the dynamical matrix .At standard temperature and pressure (STP), NiAladopts the CsCl (or B2) structure, with lattice pa-rameter 2 . ± . , giving a crystal densityof 5.912 g/cm . A plane wave cutoff of 900 eV and asymmetry-reduced 10 k -point grid was sufficient toconverge the Pulay-corrected ground states to ∼
10 meV/˚A or better. This level of convergence in pres-sure was 1-2 orders of magnitude smaller than the dis-crepancy from measured pressures introduced throughthe use of DFT. B. Isotropic compression
Isotropic compression calculations were performed todetermine the cold compression curve e c ( ρ ). The latticeparameter was varied between 2.0 and 5.0 ˚A at intervalsof 0.1 ˚A, with additional calculations at intervals of 0.05 ˚Abetween 2.5 and 3.1 ˚A and 0.01 ˚A between 2.7 and 3.0 ˚A.(Figs 1 and 2.)It is instructive to fit the electronic structure calcu-lations to the Rose functional form , which has beenfound empirically to describe the compression behaviorof many elements. A functional fit such as the Rose formis useful in that it provides a much more compact rep-resentation of the cold curve than does the tabulated re-lation obtained from a series of electronic structure cal-culations. It is interesting to assess the accuracy of thisfunctional form in reproducing the expansion region aswell as compressed states. For use with atomic struc-ture calculations, in which the binding energy containsa contribution from electrons to atoms as well as atomsto form the solid, the variation of binding energy withscaled compression can be described by e = e − e (1 + a + 0 . a ) e − a , (2)which is a slight generalization of the original form of therelation . The parameter a is a scale length defined withrespect to the Wigner-Seitz radius r W S as a function ofspecific volume, a ( v ) = r W S ( v ) − r W S ( v ) l . (3)The pressure along the cold compression curve is then p = − B ( v/v ) / − v/v ) / (1 − . a + 0 . a ) e − a . (4)The zero-pressure bulk modulus is related to the bindingenergy scale by B = e m a πr W S ( v ) l . (5)The cold curve for a material is then described by threeprincipal parameters – ρ = 1 /v , e or B , and l – andan energy offset e . The Rose function has a theoreticalconnection with simple metals, in which the compressionproperties are dominated by the electron density. Here,it was regarded as a convenient functional form for inter-polating and smoothing data.Non-linear optimization was used to obtain parametersfrom the quantum-mechanical predictions of the frozen-ion cold curve in pressure-volume or energy-volumespace. Some optimization schemes were unstable whenthe equilibrium volume v was included as a free parame-ter. In practice, v was first estimated by inspection, andthe remaining parameters ( B , l , and e ) were calculatedfor fixed v . This process was repeated for a few differ-ent values of v to investigate the sensitivity, and then arobust but inefficient Monte-Carlo optimization schemewas used to locate the optimum value of v .The Rose function was not able to reproduce the shapeof the quantum mechanical cold compression curve overits whole range. Separate fits were produced using thewhole set of quantum mechanical states, and using onlythe states under compression, which is the more relevantpart of the curve for the shock wave studies of interest.The fit to the compression portion was good: generallywithin the numerical scatter of the quantum mechani-cal states. It is possible that the Rose function performsworse in expansion because it does not represent ade-quately the localization of electrons on individual atomsas the rarefied metal ceases to conduct. The predictedparameters for NiAl were bracketed by the (observed)parameters for Ni and Al, but not so as to suggest anysystematic trend (e.g. averaging) that could be used to -1080-1075-1070-1065-1060-1055-1050-1045-1040 0 2 4 6 8 10 12 14 16 18 s pe c i f i c i n t e r na l ene r g y ( M J / k g ) mass density (g/cm )full setcompression onlyQM FIG. 1: Quantum mechanical predictions of cold compressioncurve (energy), and fit with Rose function. p r e ss u r e ( G P a ) mass density (g/cm )full setcompression onlyQM FIG. 2: Quantum mechanical predictions of cold compres-sion curve (pressure), and fit with Rose function (logarithmicscale).The absolute value of pressure is plotted in regions oftension. predict alloy properties; this highlights the inaccuracyof using mixture models as is commonly attempted topredict the properties of alloys, particularly intermetalliccompounds . (Table I and Figs 1 to 2.) C. Thermal excitation of the electrons
At elevated temperatures, excitations of the electronicstates can contribute to the free energy. The electron-thermal contribution e e ( ρ, T ) was calculated from theKohn-Sham electron band structure by populating thecalculated states according to Fermi-Dirac statistics, us- TABLE I: Rose parameters fitted to ab initio frozen-ion cold compression curve for NiAl. ρ l B e (g/cm ) (˚A) (GPa) (MJ/kg)full set 6.200 0.2860 177.8 -1069compression only 6.227 0.2966 189.0 -1068compression only, adjusted by 7 GPa 6.007 0.2934 164.8 -1070Ni 8.90 0.270 186.0 -Al 2.70 0.336 72.2 -Parameters for elemental Ni and Al are shown for comparison. ing the procedure applied previously to other elements in-cluding Si . The band structure was calculated at a tem-perature of 0 K in the frozen-ion approximation: the elec-tron states were not re-calculated self-consistently withthe state occupations at finite temperatures. Over therange of states considered, which included temperaturesup to 10000 K, the electron excitation was rather small.This can be seen by considering the electron-thermal en-ergy, found to make only a small contribution to the EOSin this regime, so any error in the band energies from thelimitations of DFT or electron-phonon coupling would bea small correction to this small contribution.The electron wavefunctions were represented at a finiteset of positions in reciprocal space, { ~k i } , reduced by sym-metry operations so states at each k -point had a weight w i . The energy levels were used directly in estimating e e ( v, T ), rather than collecting them into a numericaldistribution function g ( E ).Given the set of discrete levels for each compression,the chemical potential µ was found as a function of tem-perature T by constraining the total number of valenceelectrons N : X i w i e ( E i − µ ( T )) /k B T + 1 = N, (6)using an iterative inversion algorithm. Once µ ( T ) hadbeen determined in this way, the expectation value ofthe electronic energy was calculated: h E ( T ) i = X i E i w i e ( E i − µ ( T )) /k B T + 1 . (7)Repeating this calculation for each compression, and di-viding by the mass of the atom to obtain the specificenergy of excitation, the electron-thermal contribution e e ( ρ, T ) to the EOS was obtained.The electron-thermal energy was at least an order ofmagnitude smaller than the lattice-thermal energy overthe range of temperatures and compressions considered.At a fixed temperature, the energy decreased with com-pression as the bands broaden with respect to energy,decreasing g ( µ ). The electron-thermal energy did how-ever increase more rapidly with temperature than didthe lattice-thermal energy, so at temperatures above afew electron-volts the electron-thermal energy dominates.(Fig. 3.) D. Phonon modes
The thermal motion of the atoms and its contribu-tion to the EOS were calculated in terms of the phononmodes of the lattice, following the method described pre-viously for Si . In general, the restoring force for dis-placement each atom from equilibrium may be anhar-monic. There is a large increase in computational com-plexity in calculating derivatives in the lattice potentialenergy beyond those required for harmonic phonons. Inthe present work, we have calculated effective quasihar-monic phonon modes for relevant amplitudes of atomicdisplacement, and assessed the sensitivity of the EOS todifferent choices of displacement.Quasiharmonic phonons were calculated using a force-constant method . Ab initio elements in the dy-namical matrix were obtained by calculating the forceon all the atoms when one atom was perturbed from itsequilibrium position, from the charge distribution in thequantum mechanical description of the electron groundstate.When atom i is perturbed by some finite displacement ~u i from its equilibrium position ~r i , the force on eachatom can be found from the electron ground state. IfΦ is the electron ground state energy, the calculationprovides elements of the stiffness matrix D , between thedisplaced atom i and all other atoms j in the supercell. D ( ~r i − ~r j ) ≡ ∂ Φ ∂~u i ∂~u j ≃ ∂ Φ( α ˆ u i ) ∂~u j α . (8) ∂ Φ( α ˆ u i ) /∂~u j is the force ~f j on atom j when atom i isdisplaced by a distance α in the direction (unit vector)ˆ u i . Because partial differentiation operators commute fora smooth function, a row and column of the eigenprob-lem can be determined from an electron ground statecalculation with a displacement ~u i along one of the co-ordinate directions, by dividing the force on each atomby the displacement. Other elements can be generatedusing symmetry: for the CsCl it was sufficient to perturbthe Ni atom at (0 , ,
0) and the Al atom at ( , , ) inturn along the [100] direction in order to obtain the entirematrix of force constants.The squared phonon frequencies are then the eigenval-ues of ω ~u i = X j ∂ Φ ∂~u i ∂~u j .e i~k. ( ~r i − ~r j ) ~u i √ m i m j , (9)where m i is the mass of atom i . The matrix on the right-hand side is the dynamical matrix, ˜D :[ ˜D ( ~ k )] αβ ≡ X ij ∂ Φ ∂ [ ~u i ] α ∂ [ ~u j ] β .e i~k. ( ~r i − ~r j ) √ m i m j , (10)where square brackets are used to denote an element ofa matrix.Restoring forces were calculated and the dynamicalmatrix constructed for several magnitudes of the dis-placement of the atoms. This allowed the deviation froma harmonic potential to be estimated, and can be usedto determine some of the anharmonic components in thepotential. The displacements chosen were 0, ± . .
01 of the lattice parameter of the 2 × × a = 3 . .
9, 2 .
7, 2 .
5, 2 .
3, and 2 . ± .
005 eV/˚A. This linearityindicates that the potential surface experienced by theatoms is harmonic, an assertion which was investigatedby constructing EOS based on different magnitudes ofdisplacement.To calculate the density of phonon states g ( ω ) fora structure with a given set of lattice parameters, thephonon eigenproblem – diagonalizing ˜D ( ~k ) – was solvedfor each of a set of wavevectors ~k . These were chosenrandomly with a uniform distribution over the Brillouinzone, by choosing the components to be three indepen-dent random numbers with uniform distribution between0 and 1. As was found previously , the density of statesconverged slowly with the number of wavevectors. Wedid not attempt to calculate fully-converged densities ofstates, as previous experience showed that, as an inte-grated property of the density of states, the EOS wasonly sensitive to low moments of the distribution. Thesemoments did not change significantly with compression.The variation of lattice-thermal energy with tempera-ture was found by populating the phonon modes accord-ing to Boltzmann statistics . At a given mass density,the lattice thermal energy is E l ( T ) = X i g ( ω i )¯ hω i (cid:18) e ¯ hω i /kT − (cid:19) , (11)from which the lattice-thermal contribution e l ( v, T ) tothe EOS was found by normalizing to 3 modes per atom.(Fig. 3.)Although the phonon modes could be calculated reli-ably around zero pressure as has been found previously ,some negative eigenvalues of the dynamical matrix werefound at higher compressions, giving a small proportionof imaginary phonon frequencies. Imaginary frequencies s pe c i f i c i n t e r na l ene r g y ( M J / k g ) temperature (K)3.12.1 lattice3.1 2.1electrons FIG. 3: Thermal energies. The lattice-thermal energy wascalculated from phonon densities of states which were deducedusing different atomic displacements (solid: displacement of0.001; dashed: 0.01): 3.1, 2.9, 2.7, 2.5, 2.3, and 2.1 ˚A mov-ing up the vertical axis as the zero-point energy increaseswith compression. The electron-thermal energies were cal-culated by populating the zero-temperature band structure.Note that the electron-thermal energy decreases with com-pression, as the bands move to higher energies.Note the finite and compression-dependent value of theDebye temperature, at which all curves asymptote to thesame straight line. indicate possible instabilities in the lattice: the restoringforce on a displaced atom does not increase with displace-ment, so the displacement can grow. This situation maybe related to a major phase transition, a minor perturba-tion of the structure which is dynamically stabilized at fi-nite temperature , a restoring force that is non-linear indisplacement, or it may be caused by minor numerical in-consistencies when some force components which shouldbe identical by symmetry are calculated redundantly indifferent directions or by displacing different atoms. In-finitesimal displacements will not distinguish betweenthese possibilities, though finite displacements may beused to investigate the structure of the potential field ex-perienced by each atom. In either case, the local poten-tial is no longer quadratic, and the quasiharmonic modelis not strictly valid. Minor perturbations of the structureand non-linear restoring forces can be treated formallyby renormalization of the phonon modes , though thisprocedure is unwieldy to apply consistently with the de-tailed shape of the restoring force at finite displacements.In constructing the EOS for Si , where nonlinear con-tributions to the restoring force were more pronounced,the principal shock Hugoniot was found to be insensitiveto details of the phonon density of states, including thetreatment of imaginary frequencies. We investigated thesensitivity of the NiAl EOS to different methods for tak-ing account of the imaginary frequencies: treating themas freely translational – contributing a heat capacity of k B / E. Construction of the equation of state
For computational convenience, the zero-point energyof the lattice modes was subsumed in the lattice-thermalcontribution rather than the cold curve; thus the coldcurve strictly assumed frozen ions. This assumption doesnot affect the accuracy of the resulting EOS, but thecold curve is not physically real as defined. The electron-thermal energy can be predicted from band structure cal-culations closely related to the electron ground state cal-culations. However, we have found for many materialsthat the electron-thermal energy makes a negligible con-tribution to the EOS of the solid, and the contribution inNiAl has been calculated to be relatively unimportant ,so it was ignored in the present work.
1. Thermodynamic completion
Given the total specific energy e ( T ) along each iso-chore, the specific entropy s was found by integration ofthe second law of thermodynamics ( de = T ds − pdv ): s ( T ) = Z T dT ′ T ′ ∂e∂T ′ . (12)The (specific) free energy f was then calculated from f = e − T s , and the pressure p was calculated by differ-entiating f . The thermodynamic functions were repre-sented by tables, so local polynomials were fitted throughadjacent sets of points – generally quadratics through setsof three points – to allow differentiation and integrationto be performed.EOS were generated in SESAME 301 format . Thisconsists of rectangular tables of pressure p (GPa) andspecific internal energy e (MJ/kg) as functions of tem-perature T (K) and density ρ (g/cm ). For each of thedensities in the original cold curve calculations, stateswere calculated along an isochore from T = 0 to 10000 K.Although the most straightforward form to generate,SESAME table 301 is rather inconvenient for hydrody-namic calculations. A standard hydrocode requirementis for an EOS of the form p ( ρ, e ). To derive this from atable 301 requires T to be found by inverse interpolationgiven ρ and e , and then T and ρ used to find p . In thepresent work, bilinear interpolation was used in finding p and e at states between the ordinates of ρ and T . TABLE II: Pressure offset and bulk modulus for ab fere initio equations of state. atom offset bulk modulusdisplacement (GPa) (GPa)
2. Adjustment to reproduce STP mass density
As described previously , any EOS can be correctedto reproduce the observed STP state by adding a pres-sure offset ∆ p c and for consistency a corresponding en-ergy tilt ∆ e c = − v ∆ p c . We have found that the pressureoffset improves the agreement with compression data bet-ter than do other types of adjustment, such as scalingof the mass density. However, the quantum-mechanicalcold curves asymptoted correctly toward zero pressure asthe lattice parameter became large, so a constant pres-sure offset would not be accurate at low density. Thepressure offset was calculated from the pressure given byeach EOS at 293 K and the observed STP crystal densityof 5.912 g/cm (Table II). A modified Rose fit was cal-culated for the corrected cold curve (Table I). Of course,this fit does not reproduce the STP mass density becausethermal expansion is not included. One straightforwardtest of an EOS is to calculate the bulk modulus. For NiAl,the bulk modulus has been measured to be 156 ± and as inferred from the elastic constants is 166.0 GPa .The adjusted first principles – ab fere initio – EOS gavebulk moduli which lay close to these values (Table II). III. ISOTHERMAL COMPRESSION
Diamond-anvil measurements have been reported forthe isothermal compression of NiAl to 25 GPa . Isother-mal compression predictions were extracted directly fromthe p ( ρ, T ) table. Isotherms from the EOS based on theground state energies passed through the diamond-anvildata within its scatter. (Figs 4 and 5.) IV. SHOCK COMPRESSION
The theoretical EOS were used to predict the prin-cipal shock Hugoniot, by solving the Rankine-Hugoniotequations linking the states on either side of the shockto its velocity u s : u s = v p − p v − v (13) u p = p [( p − p )( v − v )] (14) e = e + 12 ( p + p )( v − v ) (15) c o m p r e ss i on pressure (GPa)predictiondiamond cell data FIG. 4: Comparison between calculated ambient isothermsand diamond anvil data. Alternative theoretical EOS gaveisotherms which were not significantly different. c o m p r e ss i on pressure (GPa)predictiondiamond cell data FIG. 5: Comparison between calculated ambient isothermsand diamond anvil data (detail at lower pressures). where subscript ‘0’ denotes material ahead of the shock(with u p = 0). Given the EOS in the form p ( v, e ) thisset of equations can be closed, allowing the Hugoniot(locus of states reached from the initial state by a singleshock) to be calculated. The phonon and electron modeswere populated at each mass density and temperature inthe construction of the EOS, so the Rankine-Hugoniotequations were in effect solved self-consistently with thepopulation of the thermal modes, though this was doneindirectly through the use of the EOS.Like the EOS itself, the shock Hugoniot was calculatedup to pressures of several hundred gigapascals. The com-parison with isothermal compression data demonstratesthe accuracy of the EOS, and the electron band-structureremains valid until the pseudopotentials start to over- p r e ss u r e ( G P a ) mass density (g/cm )0.0010.001, rescaled modes0.01 FIG. 6: Comparison between principal shock Hugoniot fromdifferent phonon treatments, in density – pressure space. Theatom displacement is expressed with respect to the superlat-tice parameter. lap, which typically requires pressures in the terapascalregime. As well as providing an a priori prediction ofthe EOS and Hugoniot, the high-pressure calculationsserve to explore the sensitivity of the theoretical predic-tions to different assumptions and simplifications used inconstructing EOS, and therefore where greater care – orexperiment – is needed to constrain the EOS.The quasiharmonic EOS showed some sensitivity tothe atom displacement and the treatment of the imag-inary modes. The variation is an indication of the un-certainty in the theoretical EOS, and of regimes in whichanharmonic terms in the lattice-thermal energy (phonon-phonon interactions) may contribute significantly. Inpressure-density space, the Hugoniots were very similarbelow around 80 GPa, then differed by around 15% athigher pressures. In shock speed-particle speed space,the Hugoniots varied by 5% at low pressures, became co-incident for shock speeds around 7 km/s (60 GPa), thendeviated by around 5% at higher pressures. The devia-tion was most pronounced in pressure-temperature space:around 15%. With the density of phonon states correctedto remove imaginary modes – labeled as ‘rescaled’ in thefigures – the principal Hugoniot for a displacement of0.001 lay much closer to that for 0.01. (Figs 6 to 8.)The EOS showed little sensitivity to the inclusion ofthe electron-thermal contribution, except in pressure-temperature space where it rose linearly with temper-ature, reaching around 10% in pressure by 10000 K. Theexaggerated effect in temperature is not surprising: overthe regime studied the EOS is dominated by the coldcompression curve, so a given, relatively small, differencein thermal pressure equates to a considerably larger dif-ference in temperature. (Figs 9 to 11.) s ho ck s peed ( k m / s ) particle speed (km/s)0.0010.001, rescaled modes0.01 FIG. 7: Comparison between principal shock Hugoniot fromdifferent phonon treatments, in particle speed – shock speedspace. The atom displacement is expressed with respect tothe superlattice parameter. p r e ss u r e ( G P a ) temperature (K) 0.0010.001, rescaled modes0.01 FIG. 8: Comparison between principal shock Hugoniot fromdifferent phonon treatments, in temperature – pressure space.The atom displacement is expressed with respect to the su-perlattice parameter.
V. SHOCK WAVE EXPERIMENTS
The experiments described here were intended to helpvalidate the theoretical EOS and to calibrate the modelof flow stress. In many cases the samples were recoveredafter the experiments, so metallographic analysis couldbe performed in future if desired.To generate shock states by impact, the TRIDENTlaser was used to launch flyer plates, each of which thenimpacted a stationary target. Two series of experimentswere performed. In the first, the flyer was Cu and thetarget comprised a crystal of NiAl, sometimes releasing p r e ss u r e ( G P a ) mass density (g/cm )e e ignoredT=0 band structure FIG. 9: Comparison between principal shock Hugoniot fromdifferent electron-thermal treatments, in density – pressurespace. The phonons were calculated with a displacement of0.001 times the superlattice parameter, without rescaling. s ho ck s peed ( k m / s ) particle speed (km/s)e e ignoredT=0 band structure FIG. 10: Comparison between principal shock Hugoniot fromdifferent electron-thermal treatments, in particle speed –shock speed space. The phonons were calculated with a dis-placement of 0.001 times the superlattice parameter, withoutrescaling. into a polymethyl methacrylate (PMMA) window. In thesecond series, the flyer was NiAl and the target a trans-parent LiF crystal. The flyer speed and surface velocityof the sample were measured by laser Doppler velocime-try.The laser pulse was 600 ns long, and the flyers werearound 50 to 400 µ m thick. Velocities were a few hundredmeters per second with a laser energy of 5 to 20 J over aspot 5 mm in diameter.The accuracy of the laser flyer technique was previ-ously evaluated in experiments on the EOS of Cu , and p r e ss u r e ( G P a ) temperature (K)e e ignoredT=0 band structure FIG. 11: Comparison between principal shock Hugoniot fromdifferent electron-thermal treatments, in temperature – pres-sure space. The phonons were calculated with a displacementof 0.001 times the superlattice parameter, without rescaling. has also been used to measure the EOS of NiTi . Theexperiments reported here were performed at TRIDENTas part of the ‘Pink Flamingo’ (December 2001) and ‘Fly-ing Pig’ (March 2002) campaigns. A. Sample preparation
Single crystal samples of NiAl were grown from themelt using the optical floating zone technique. Feed-rodswere prepared via arc melting, with excess Al includedin the initial charge to compensate for losses during rodpreparation and crystal growth. Crystals were grown at15 mm/hr along h i , starting from an oriented seed. Tocompensate for the greater evaporation rate of Al, theinitial melt was prepared with a slight excess of Al. It isdifficult to predict the evaporation rate precisely, and thecrystals were slightly Al-rich. Single crystal NiAl was alsoobtained from the General Electric Corp; this materialwas closer to stoichiometry. The orientation was deter-mined by back-reflection Laue diffraction. Samples weresliced parallel to (100) and (110) planes, then ground andpolished to the desired thickness using diamond media,to a 1 µ m mirror finish. The grinding process impartedsome pre-strain close to the polished surfaces. B. Experimental configurations
Several different configurations of flyer experimentwere used. In all cases, the flyer was attached to itstransparent substrate and spaced off from the target as-sembly by a ‘barrel’ comprising a stack of plastic shims.The barrel was typically around 500 µ m long, allowing assembly VISAR f l ye r barrel target (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) substrate beamslaserdrive FIG. 12: Schematic cross-section of laser-launched flyer im-pact experiments. enough space for the flyer to accelerate before impact.(Fig. 12).In the initial experiments with a Cu flyer and NiAltarget, the sample covered only half of the flyer, so theflyer could be seen and its velocity measured. Severalvariants were used in the design of the target assembly,to investigate the accuracy of data from each. In most ofthese experiments, the NiAl sample was in contact witha PMMA release window; in some of the PMMA win-dow experiments, the window was stepped to provide amore accurate measurement of the time at which impactoccurred and thus the shock transit time through thesample. Experiments were also performed in which thesample was mounted on a Cu baseplate: the baseplateobscured the view of the flyer, but shock breakout at thesurface of the baseplate provided a measurement of thepressure (from the peak free surface particle speed) andgave a relatively accurate measurement of the time atwhich the shock entered the sample. The baseplate de-sign also avoided difficulties from light reflected from thefree surface of the PMMA windows, which sometimesobscured the signal from the flyer or the sample. Thebaseplate had to be relatively thin to avoid decay of theshock, and the sample was not recovered from experi-ments using this design. (Fig. 13.)The flyer speeds were low compared with the soundspeed in the target materials, so the finite accuracy ofassembly – several micrometers in co-planarity or thick-ness – and the finite temporal registration between thepoint and line VISARs resulted in large uncertainties inshock transit times. The second series of experiments wasdesigned to minimize the uncertainty in inferred particlespeed and shock pressure: by using the deceleration of aNiAl flyer with a window of known properties (LiF), theHugoniot state could be inferred from the point VISAR0 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) window PMMAwindowNiAlsample NiAlsample CubaseplateNiAlsample (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
PMMA
FIG. 13: Schematic of target assembly for PMMA release andbaseplate configurations. Center: stepped window providingbetter measurement of impact time. In all cases the flyerapproaches from the left. record only, the less precise line VISAR being used as acoarse velocity measurement (e.g. to help count fringejumps in the point VISAR record) and to verify flatness.
C. Target assembly
PMMA substrates were used, coated with layers of C,Al, and Al O , in order to absorb the laser energy andinsulate the flyer from heating . Cu flyers were punchedfrom foils purchased from Goodfellow Corp. The foils hadstriations and machining marks which could generate in-terference patterns that could confuse the interpretationof the laser velocimetry records. The foils were polishedmanually using diamond paste to reduce the regularity ofthe marks. The material for the NiAl flyers was roughlytriangular in shape. Rather than cutting or punchingflyer disks before assembly, the triangles were attachedwhole to the substrates, and the drive laser punched outthe central 5 mm to form the flyer. The material re-maining attached to the substrate formed a seal whichprevented plasma from the drive escaping radially. Thebarrel was much shorter than the flyer diameter, so theattachment of the edges did not make the central por-tion of the flyer curve measurably and did not reduce itsspeed.Components were glued together with five-minuteepoxy. Each flyer was clamped to its substrate whilethe glue set, to minimize the thickness of the glue layer.The initial viscosity of the glue was low, so the glue thick-ness was estimated to be negligible, essentially filling insurface irregularities. Components of the target assem-bly were clamped, and glued together by small drops attheir corners or edges to avoid introducing any layer ofglue between components which would change the shockand optical properties. (Figs 14 and 15.) substrate impactfiducialwindowsampleflyer on spacer FIG. 14: Example of components of shock experiment duringassembly (stepped release window design). The flyer is 5 mmin diameter; the sample is visible through the window. fiducialwindowflyer(throughwindow) target
FIG. 15: Example of assembled components (stepped releasewindow design).
D. Diagnostics
Laser Doppler velocimetry of the ‘velocity interferom-etry for surfaces of any reflectivity’ (VISAR) type wasused to measure the velocity histories of the flyer andthe sample. A point-VISAR and a line-imaging VISARwere used simultaneously, the point-VISAR signal beingrecorded on digitizing oscilloscopes and the line-VISARsignal on an optical streak camera. The point VISAR op-erated at a wavelength of 532 nm with a continuous-wavesource, and the line VISAR at a wavelength of 660 nmwith a pulsed source ∼ µ s long. Timing markers wereincorporated on the streak record, at intervals of 200 ns,to provide a temporal fiducial and to allow non-linearitiesin camera sweep to be removed. Relative timing of thepoint and line VISARs was deduced by comparing theposition at which a shock wave appeared in a flyer im-pact experiment; the relative timing had an uncertaintyof ∼ E. Drive beam
The TRIDENT laser was operated in long-pulse mode,as in the previous flyer work , using an acousto-optical modulator to reduce the rate at which the pulseintensity rose and to control its shape. The drive pulsewas chosen to be ∼
600 ns long (full width, half maxi-mum), and was delivered at the fundamental wavelengthof the laser: 1054 nm (infra-red). The pulses generatedwere asymmetric in time, with a long tail.An infra-red random-phase plate (RPP) was added tosmooth the beam; this made a significant difference tothe spatial uniformity. The beam optics were arranged togive a spot 5 mm in diameter on the substrate. The driveenergy was quite low, so no RPP shield was included.The RPP collected a small amount of debris from thesubstrate, but was not significantly damaged.
F. Results
Eight experiments were performed with Cu flyers im-pacting NiAl, and seven with NiAl flyers impacting LiFwindows (Table III). On one shot (14138), poor fringecontrast meant that the reference velocity could not bemeasured by velocimetry, so it was deduced from the laserenergy instead.The drive energy was measured with a calorimeter, andthe irradiance history of the drive pulse with a photodi-ode. The uncertainty in energy was of the order of 1 J.The pulse shape was repeatable at the same and differentenergies.The VISAR records were used to measure the velocityhistory and flatness of each flyer. The flyers were flat towithin the accuracy of the data over the central 3 mm,with a slight lag at the edges. In some experiments, werecorded the impact of the flyer with a window and thuswere able to measure the flatness directly after severalhundred microns of flight. Most of the flyers were stillaccelerating slightly at the end of the record. There wasevidence of ringing during acceleration, but no sign ofshock formation or spall in the flyers.In the release window and baseplate experiments, thevelocity history at the surface of the NiAl generally ex-hibited a precursor to a particle speed of 15-30 m/s aheadof the main shock wave. This precursor was presumablyan elastic wave. The rising part of the shock generally ex-hibited some structure, consistent with reverberations ofthe elastic wave between the surface and the approachingshock. The peak velocity was followed by deceleration of25-30 m/s (with some outliers) and reverberations, con-sistent with spallation and ringing. (Figs 16 to 18.)In the window impact experiments, the flyer speed justbefore and just after impact were determined from theVISAR record (Fig. 19). v e l o c i t y ( m / s ) time (ns) sample(point VISAR)flyer(line VISAR) FIG. 16: Example velocity history from a window release ex-periment (shot 14128). v e l o c i t y ( m / s ) time (ns) FIG. 17: Example spatially-resolved velocity history from awindow release experiment (shot 14140: stepped release win-dow).The velocity from successive fringes has been displaced by100 m/s for clarity.
G. Analysis
The impact of a Cu flyer onto a NiAl target, releasinginto a window, allowed a point on the principal Hugo-niot to be deduced, relying on the assumption that theNiAl release isentrope through the shock state gener-ated by the impact was equal to the principal Hugoniot(Fig. 20). At the pressures of a few gigapascals gener-ated in these experiments, this assumption is accurate toa percent or better. This analysis is simplest for releaseinto vacuum, where the particle speed in the sample canbe estimated as half of the free surface speed. The dif-ference between this and the flyer speed is equal to the2
TABLE III: Flyer impact experiments.
Shot NiAl Driver Targetorientation thickness speed thickness speed ( µ m) (m/s) ( µ m) (m/s) release window ±
15 200 331 ± ±
20 217 390 ± ±
10 217 232 ± baseplate ±
10 398 208 ± ±
10 398 260 ± ±
15 398 195 ± stepped disk ± ± ± ± window impact ± ± ± ± ± ± ± ± ± ± ± ± ± ± µ m copper. The driver speed is the peak free surface speed of the baseplate, and the flyer speed onimpact otherwise. The impact window was a LiF crystal, 2 mm thick, with (100) planes parallel to the impact surface. The‘target speed’ in the window impact experiments is the speed of the interface between the NiAl flyer and the LiF windowimmediately after impact. v e l o c i t y ( m / s ) time (ns) FIG. 18: Example velocity history at the free surface of thesample in a baseplate experiment (shot 14136). particle speed in the flyer, and the Hugoniot of the flyermaterial can be used to find the shock pressure. Refer-ence Hugoniots were calculated from published EOS .Hugoniot states were obtained, with pressures between 2and 8 GPa. States deduced by window release and freesurface velocity were consistent. The principal Hugoniotfrom the ab fere initio EOS passed as closely through the -50 0 50 100 150 200 250 300 350 0 200 400 600 800 1000 1200 1400 1600 1800 v e l o c i t y ( m / s ) time (ns)1438114388 impact FIG. 19: Example velocity histories from window impact ex-periments. data as any unique line is likely to. (Fig. 21.)A complication of the window impact experimentscompared with the Cu flyer experiments was that thestress state at the interface between the sample and thewindow depends more sensitively on the elastic strain inboth components. While the elastic constants may beknown, the flow stress (or yield strength) is not known3 (unknown)particle speed p r ess u r e windowHugoniot(known)flyer speed(measured)interface speed(measured)targetisentrope(unknown)shock stateon impactrelease stateflyerHugoniot(known)targetHugoniot FIG. 20: Schematic of construction used to deduce Hugoniotpoint from window release data with a flyer of known material. p r e ss u r e ( G P a ) particle speed (km/s)window releasefree surfaceQM EOS FIG. 21: NiAl Hugoniot points deduced from the impact ofCu flyers, compared with the ab initio equation of state. for all time scales. This was not such a concern for theCu flyer experiments because the flow stress of Cu andPMMA are much lower than for LiF . Calculations weremade using different assumptions about the flow stress,to assess the uncertainty and the likely behavior of LiFin these experiments.Neglecting elasticity, a measurement of the particlespeed immediately before and after impact ( u and u )can be used to determine a state on the principal Hugo-niot of the flyer with reference to the principal Hugoniot no r m a l s t r e ss ( G P a ) particle speed (km/s)QM EOS(100) measurements(110) measurements FIG. 22: Hugoniot points deduced for nickel – aluminum alloy,compared with the ab initio equations of state. of the window, which must be known. Immediately afterimpact – for a time dictated mainly by the shock andrelease transit time through the flyer and window – thematerial in both components at the impact surface is atthe same pressure and traveling at the same speed. TheHugoniots are expressed in pressure – particle speed ( p – u p ) space. In the shocked region, the particle velocitywith respect to the undisturbed material in the window is u , and in the flyer u − u . As the same pressure exists inboth materials, and it can be calculated from the Hugo-niot of the window material p W ( u ), a Hugoniot statecan be determined in the flyer: ( u − u , p W ( u )). TheRankine-Hugoniot relations can then be used to calculatethe other state parameters: mass density ρ , shock speed u s , and the change in specific internal energy e (Table IVand Fig. 22).More rigorously, the materials at the impact interfaceare at the same state of normal stress s rather thanpressure, and under uniaxial compression. By the samearguments as above, if the shock Hugoniot of the window– now considering its response as a stress tensor s , at leastthe normal component – is known, then the normal stressof the flyer material is thus also known. The stress com-ponents for a solid depend on the elastic strain and theelastic constants. Elastic strain can be relieved by plas-tic flow, which limits the deviation of normal stress fromthe hydrostatic pressure. The elastic constants of com-mon window materials are well-known, but the finite rateof plastic flow – which can also be considered in termsof a flow stress or yield stress which varies with strainrate – means that experiments exploring the responseon different time scales may induce significantly differentdegrees of elastic strain for the same compression. Ourexperiments used flyers which were relatively thin com-pared with previous experiments, so it is possible thatthe elastic strains were somewhat greater.Another aspect of this complication is that we ide-4ally want to determine the EOS. Data from window im-pact experiments comprise the EOS plus the elastic shearstress. For a single experimental measurement in isola-tion, there is no way to separate the hydrostatic pressure(EOS) from the elastic stress, but some values can be ex-tracted given data from several experiments at differentpressure, and/or knowledge of the EOS or elastic andplastic behavior. In the present case, relevant comple-mentary data include the STP elastic constants of NiAl,our QM EOS, and low strain-rate measurements of theflow stress. We considered the limiting case of no plas-tic flow (i.e. maximum elastic stress for a given uniaxialcompression), and also cases with varying flow stresses inthe window and the NiAl flyer.For predicted shock Hugoniots including elasticity, theelastic stress was added to the Hugoniot pressure cal-culated from the EOS. For each state on the Hugoniot,the uniaxial compression η = ρ/ρ was used to calculatethe strain tensor (using the Green-St Venant finite strainmeasure ) e = (cid:0) /η − (cid:1) (16)and hence the strain deviator ǫ = (cid:0) − /η (cid:1)
00 0 . (17)The corresponding elastic stress is given by ∂ [ s ] ij ∂ [ e ] kl = [ c ( ρ )] ijkl (18)where c is the tensor of elastic constants, strictly a func-tion of e too. In deviatoric form, s ≡ σ ( ρ, ǫ ) − p ( ρ ) I (19)where p ( ρ ) is the EOS and ∂ [ σ ] ij ∂ [ ǫ ] kl = [ c ( ρ )] ijkl . (20)Plastic flow acts to limit the components of the elasticstrain deviator ǫ ; in common practice it may be expressedas a constraint on the magnitude of the stress deviator.The EOS is a more general representation of the bulkmodulus B = 13 ( c + 2 c ) . (21)Comparing the response of a crystal to uniaxial compres-sion along [100], with stiffness to uniaxial compression c , we can define a shear modulus µ = 12 ( c − c ) . (22) no r m a l s t r e ss ( G P a ) particle speed (km/s)hydrodynamicelasticelastic-plastic FIG. 23: Stress states deduced for NiAl, using different as-sumptions about plasticity in the LiF window.
Using the published STP elastic constants for NiAl ( c =211 . c = 143 . c = 112 . , µ =34 . Y chosen to improve the match to theexperimentally-measured shock states (Table IV). If theLiF window was treated as purely elastic then no treat-ment of the NiAl reproduced the experimental states.If the LiF was treated as elastic-plastic with flow stress Y = 0 .
36 GPa as deduced from gas gun experiments, thenthe measured states were reproduced fairly well with aflow stress of 0.53 GPa in the NiAl. (Figs 24 and 25, alsoshowing shock states deduced from the Cu flyer exper-iments. The Cu flyer data were not corrected for elas-tic response in the window: the correction is smaller forPMMA.)The precursor waves and the deceleration following thepeak of the shock can be used to infer flow stresses andspall strengths. However, simple models parameterizedfrom these data can be misleading, so the detailed con-stitutive behavior of NiAl will be reported separately inthe context of microstructural response. The true (i.e.normal) stress to induce plastic flow at lower strain rateshas been reported at around 1.5 GPa , corresponding toa flow stress of around 0.4 GPa. One would expect thestresses to be at least as large on the shorter time scalesof the laser experiments, so the flow stress inferred fromthe window impact experiments is quite plausible.5 TABLE IV: Shock Hugoniot states deduced from window impact experiments, using different constitutive models for LiF andNiAl. shot particle speed (m/s) normal stress (GPa)shot for different LiF models for different NiAl modelshydrodynamic Y = 0 . GPa hydrodynamic elastic Y = 0 . GPa(100) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± (110) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± no r m a l s t r e ss ( G P a ) particle speed (km/s)no NiAl strengthelastic NiAlCu flyer experimentselastic LiFplastic LiF FIG. 24: Hugoniot points deduced for NiAl, compared withthe ab initio equation of state. Points with large error barsare from the Cu flyer experiments.
VI. CONCLUSIONS
The frozen-ion compression curve for NiAl in the CsClstructure was predicted using ab initio quantum mechan-ical calculations of the electron band structure. The cal-culations reproduced the lattice parameter at p = 0 to ∼
1% with no corrections, equivalent to a discrepancy ∼ ab initio pressure-volume relation cal-culated using the Hellmann-Feynman theorem was notperfectly consistent with the ab initio energy-volume re-lation; this reflects the lower precision of the stress cal-culations. The Rose functional form, found to fit theenergy-volume relation for compression of a wide rangeof elements, was found to fit the energy-volume rela-tion for compression of NiAl, but deviated significantlyin expansion. The electron band structure was used topredict electron-thermal excitations. The mechanical re- no r m a l s t r e ss ( G P a ) particle speed (km/s) 0.53 GPa1.33 GPano NiAl strengthelastic NiAlplastic NiAlCu flyer experimentsplastic LiF FIG. 25: Hugoniot points deduced for NiAl, compared withthe ab initio equation of state. Points with large error barsare from the Cu flyer experiments. sponse in the equation of state and shock Hugoniot didnot alter much when the electron-thermal contributionwas included, though the Hugoniot pressure-temperaturerelation varied by around 1% per thousand kelvin. Thecharge distribution in the electron ground states was usedto predict ab initio phonon modes, from the forces on theatoms when one was displaced from equilibrium. Thephonon density of states was quite sensitive to the mag-nitude of the displacement – though the restoring forceon the displaced atom itself suggested that the effec-tive potential it experienced was essentially harmonic –but thermodynamically-complete equations of state con-structed using the phonon modes were not sensitive to thedetails of the density of phonon states up to ∼
100 GPa.The quasiharmonic equations of state reproduced pub-lished measurements of isothermal compression of NiAlextremely well, and were also consistent with measure-6ments of the shock compression.Laser-driven flyer experiments were performed to mea-sure states on the principal shock Hugoniot of NiAl, byimpacting Cu flyers into NiAl targets and by impactingNiAl flyers against LiF windows. Shock transit timeswere not measured accurately enough to constrain theEOS. Shock states from the Cu flyer data were obtainedwith respect to the Hugoniot of Cu or PMMA. Thesestates were consistent with the ab initio
EOS for NiAl.Detailed interpretation of the NiAl flyer data depends onthe elastic-plastic behavior of the NiAl and the LiF. Ig-noring these elastic contributions, the shock states wereconsistent with theoretical EOS. If the LiF was assumedto response elastically, the stress states deduced were im-plausibly high. If it was assumed to response with thesame flow stress as observed in gas gun experiments –which typically explore somewhat longer time scales –then it was possible to reproduce the shock states bytaking either EOS and adjusting the flow stress in theNiAl. Reasonable agreement was obtained for the quasi-harmonic EOS with a flow stress of 0.53 GPa, which isconsistent with the range of values deduced from theamplitude of the elastic precursor wave in other shockloading experiments. The quasiharmonic EOS – whichwas found to reproduce quasistatic compression data ex-tremely well – is thus consistent with the shock measure- ments.
Acknowledgments
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A192/193 , pp 249-54 (1995). s pe c i f i c i n t e r na l ene r g y ( M J / k g ))
A192/193 , pp 249-54 (1995). s pe c i f i c i n t e r na l ene r g y ( M J / k g )) mass density (g/cm )pressure, fullpressure, compressionenergy, fullenergy, compressionQM0.01 0.1 1 10 100 1000 10000 0 2 4 6 8 10 12 14 16 18 p r e ss u r e ( G P a ) mass density (g/cm )pressure, fullpressure, compressionenergy, fullenergy, compressionQM0.01 0.1 1 10 10 100 1000 10000 s pe c i f i c i n t e r na l ene r g y ( M J / k g ))