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114 Los Alamos Science Number 28 2003

Shock Compression Techniques for Developing Multiphase

Equations of State

Robert S. Hixson, George T. Gray, and Dennis B. Hayes

An airplane accidentally hits a mountainside, a bird gets sucked into the turbineblades of a jet engine, a meteorite strikes a satellite or a planet, explosives blast acavity in the earth for a building foundation, a hammer strikes metal to forge a

new part, a blast wave from a nuclear explosion strikes objects in its path—in both man-made and natural settings, shock waves and impacts produce strong

impulsive loading or sudden increases in external stress. In eachcase, the scientist or engineer would like to predict the response

of the material to that dynamic loading. How much does thedensity increase? Does the material heat up? Does it melt?

The key to answering some of thesequestions lies in knowing those intrin-sic properties of materials character-ized by their equations of state. Likethe familiar equation of state (EOS) ofan ideal gas, PV = nRT, the equationof state of a material specifies a defi-nite relationship between three ther-modynamic variables, pressure P,temperature T, and volume V (or den-sity ρ = mass/V). Thus, it is not possi-ble to adjust the three variablesindependently. Rather, if the densityand temperature of a material arefixed, for example, then its pressure(as well as energy, entropy, and allother thermodynamic quantities) is aunique value determined by its EOS.

Los Alamos scientists have a longtradition of using dynamic loadingtechniques to develop the equations ofstate that describe solids and liquids atextremely high temperatures and pres-sures. During the Manhattan Project,Hans Bethe, Geoffrey I. Taylor, CyrilS. Smith, and others developed seminaltheories of material response to shockwave compression. After World War II,experimentalists Stanley Minshall,John M. Walsh, Robert G. McQueen,and others developed plate impacttechniques to make much more precisemeasurements of equations of state.Weapon designers use those equationsof state, as well as others developedmore recently, to improve the fidelityof their large-scale computer simula-tions of nuclear weapon designs.

Today the goal is to perform high-fidelity simulations that predict nuclearweapon performance and safety undera wide variety of scenarios. To achievethe required level of confidence,weapon designers need equations ofstate that faithfully account for thecomplex behavior of plutonium, urani-um, and many other metals when theyare dynamically compressed.

Under dynamic loading, metals canchange not only from solid to liquidand liquid to vapor but also from onesolid crystal structure to another. Here

we describe how we are using newshock-compression techniques involv-ing the preheating of materials to mapout the boundaries between solid–solidand liquid–solid phases. We also out-line how we use that information toconstruct sophisticated, semiempiricalmultiphase equations of state fromwhich we can predict responses ofmaterials in complex geometries,responses that have not been or cannotbe directly measured experimentally.These equations of state help with ourown interpretation of experiments andcontribute to the development of other,more comprehensive equations of statefor use in weapon design codes.

Zirconium Phase Diagram

To illustrate the development ofan EOS, we will consider our workon the EOS of zirconium, a heavyelement between titanium and hafni-um in the group 4B metals. Figure 1,the phase diagram of zirconium,shows current best estimates for thephase boundaries of zirconium in thepressure–temperature (P–T) plane. Ata constant pressure of 1 atmosphere,zirconium, which is a hexagonal,close-packed (hcp) structure at roomtemperature, will change to a less-dense, body-centered-cubic (bcc)structure when heated above 1136

Number 28 2003 Los Alamos Science 115

Multiphase Equations of State

2500

2000

1500

1000

500

00 20 40

P (GPa)

T (

K)

60 80

Liquid Hugoniot

Isotherm

α

ω

β

Figure 1. Zirconium Phase DiagramThis diagram by Carl Greeff of Los Alamos shows current best estimates for thephase boundaries of zirconium in the pressure–temperature (P–T) plane. The esti-mates are based on shock compression work and analyses from the literature.Boundaries separate the three solid phases—α (hcp), ω (hex), and β (bcc)—and theliquid phase. Each P–T point along a phase boundary defines a state in which mix-tures of the two phases can coexist in equilibrium. The locus of P–T states in whicha liquid and a solid phase can coexist is called the coexistence curve, the meltcurve, or the phase boundary. The locus of states in which two solid phases cancoexist is called a coexistence curve or phase boundary. Two coexistence curvesintersect at a “triple point,” and at that temperature and pressure, three phases cancoexist. Also shown is the principal Hugoniot, a locus of end states reached byshock compression starting from room temperature and pressure.

kelvins. At still higher temperatures,it melts. Similarly, if the metal iskept at room temperature, the hcpphase of zirconium will transform toa new, higher-density hexagonalphase when the pressure is increased.(Note that for some metals, the high-pressure phase is less dense than thelow-pressure phase.)

Using Shock Waves toDevelop Equations of State

Also shown in Figure 1 is the“principal” Hugoniot1 for zirconium.This dashed curve is the locus of endstates that can be reached throughshock wave compression. TheHugoniot rises steeply in the P–T

plane, whereas an isotherm, a locusof states reached through static com-pression (slow increase in pressure)with temperature held constant, is bydefinition, flat in the P–T plane. Boththe Hugoniot and the isotherm areuseful in developing an EOS for amaterial, but they must be combinedwith other thermodynamic informa-tion, such as the material’s heatcapacity. In general, phase diagramsand properties of the pure phases canbe experimentally determined witheither static or dynamic techniques.Results may differ because of the dif-ferences in the experimental timescale. Sometimes, those differencesare explained by time-dependentequilibration, but other differencesreflect the fact that a materialresponds differently to high strainrates than it does to low strain rates.

For the Hugoniot shown in

Figure 1, the temperatures were notmeasured directly but were calculatedfrom the EOS that we constructedusing the Los Alamos and Russiandata displayed in Figure 2. This rep-resentation of the shock Hugoniot is aplot of shock velocity (Us) versusparticle velocity (Up). Each cusp, orsudden change in slope, signals thelocation of a solid–solid phase bound-ary. The cusp forms because theshock pressure is sufficient to inducea phase change, and the resultingdensity change causes the shockvelocity to change and the singleshock to split into two shocks, onefollowing the other.

Figure 3 describes how we usetime-resolved laser-interferometrictechniques to locate the position ofthat cusp, or solid–solid phasechange, in the P–T plane. Work isunder way to use different initial tem-peratures of the material and therebymap out all the points on the phaseboundaries in Figure 1.

The gas gun facility where we per-form the experiments is shown inFigure 3(a). The gas gun acceleratesa projectile carrying a thin impactortoward a flat, very thin zirconium tar-get held in place by its edges—referto Figure 3(b). At the back of the tar-get is a laser probe connected byfiber-optic cables to a VISAR2

(velocity interferometer system for anyreflector)—refer to Figure 3(c). Bydetermining the frequency of the laserlight reflected from the center of thetarget’s back surface, the VISARdetermines the velocity history ofthat surface after impact. Asexplained in Figure 3(d), the velocityhistory of the back surface directlyreflects the structure of the shockwaves that have propagated throughthe sample and, in turn, the effects of



Figure 2. The Zirconium Shock Hugoniot The shock Hugoniot for zirconium, a plot of shock velocity (Us) vs particle velocity

(Up), was determined from Los Alamos and Russian shock-compression measure-

ments and from calculations (dashed curves) using the multiphase model devel-

oped by Greeff. The cusps in the data indicate that successively higher-impact

stresses bring the metal to final states with different crystalline phases: the α (hcp),ω (hex), or β (bcc) phase. This shock Hugoniot was used to calculate the P–T

Hugoniot displayed in Figure 1.

3

4

5

6

7

8

0 1 2 3 4

Up (km/s)

Us

(km

/s)

Los Alamos data

Russian data

Calculation for α-phase only

Calculation for three-phase model

1 This curve is named after the nineteenthcentury French scientist.

2 The VISAR was invented around 1970by Lynn Baker at Sandia NationalLaboratories. It is the tool of choiceworldwide for shockwave work.



(a) Both photo and schematic (inset) show a gas gun facility atAncho Canyon, where we perform shock compression experi-ments. Typically, a projectile is accelerated by compressed gas,or sometimes by gunpowder, down a long barrel (10 to 40 ftlong) and impacts a stationary target at speeds of 0.1 to 8km/s. (b) In our equation-of-state experiments, a flat, thinimpactor carried by the projectile strikes a flat, very thin zirco-nium sample and sends a compression wave through that tar-get. A laser probe at the back of the target focuses laser lightfrom one optical fiber onto a spot at the center of the zirco-nium plate. The light reflected from the moving surface isfocused onto a second optical fiber that leads back to a laser-velocity interferometer, called a VISAR, located in the recordingroom (c). The VISAR determines the reflected light’s frequencyshift as a function of time and thus the velocity history of theback surface of the zirconium sample. The wafer is so thin and

the aspect ratio so large (say, 40 to 1) that measurements takeless than 2 µs following impact, before any edge effects couldtravel to the target center and affect the measurements. Thistechnique determines the time-resolved velocity history of amoving surface with the times accurate to ± 1 ns and thevelocities accurate to ± 1%. (d) The shape of the surface veloc-ity history reflects the propagation of the shock wave througha multiphase material. Between 0 and 1, an elastic wavereaches the target’s back surface; between 1 and 2, a plastic(deforming) wave called P1 increases the pressure to the point2 where the material begins to change phase. Between 2 and 3,a second plastic wave (P2) increases the pressure slowly asmore of the material changes phase until the peak load isreached at 3. Finally, at 4, a trailing release (rarefaction) wave,initiated by reflection of the shock wave from the impactor’sback surface, arrives at the target’s back surface.

BreechBarrel

Catchtank

TargetProjectile

(a)

Aluminum or magnesium

Projectile Target assembly VISAR probe

Foam cushionImpactor

Timing pins

VISARprobelens

Aluminum holder

Target

Window

Input fromlaser

Optical

fibers

Output toVISAR

1Elastic

P1

0

1 Elastic to plastic transition2 Break at phase boundary3 Peak load4 Arrival of trailing release wave

2

3 4

Time

Free

sur

face

vel

ocity

P2

Release wave

(b) (d)

(c)

Figure 3. VISAR Measurements of Shock Wave Structure

shock compression on the material. Inparticular, if shock compression caus-es the material to change to a denserphase (for example, to cross the α–ωboundary or the ω–β boundary inFigure 1), the shock wave is unstableand breaks into two shock waves, ordeforming/plastic waves, called P1and P2. The first deforming wave P1has a pressure corresponding to thestress at which the principal Hugoniot

intersects the phase boundary inFigure 1. It brings the material topoint 2 in Figure 3(d). The second,slower plastic wave P2 brings thematerial to its ultimate loading stressshown as point 3 in Figure 3(d). Thedetailed shape of the transmitted com-pression wave, therefore, containsinformation about the location of onepoint on the phase boundary. Thesmall initial elastic wave (between

point 0 and point 1) complicatesinterpretation of this record and isdiscussed below.

By changing the initial tempera-ture of the sample, we can shift thestarting point of the Hugoniot curveso that it will cross the phase bound-aries at a different longitudinal stressin the material and thereby map outall the points on the phase bound-aries. Such techniques for preheatedshock compression are currentlybeing developed at Los Alamos.


Development. After locating theboundaries between different phasesof a material, we use that informa-tion to help develop sophisticated,thermodynamically consistent equa-tions of state that take into accountsome or all the possible structures ofthe material. We then use these mul-tiphase equations of state to do high-fidelity computer simulations ofexperiments that involve dynamicloading.

We model the EOS for each purephase with, for example, a semiem-pirical, analytical form for theHelmholtz free energy (HFE) anddetermine the parameters in themodel from shock wave experiments,isothermal compression data, and anyother available data. To get a thermo-dynamically complete HFE in themixed-phase region, we must alsospecify the entropy and energy at onereference point. We know that twophases have the same Gibbs freeenergy (GFE) along their coexistencecurve. Therefore, by requiring thatthe calculated coexistence curves(that is, the points at which the calcu-lated GFE for the two phases areequal) match the measured phaseboundaries, we can determineuniquely all the pure-phase referenceenergies and entropies.



0.35

0.30

0.25

0.20

0.15

0.10

0.5

0

Up

(km

/s)

t (µs)2.5 3.0 3.5

1

2

3 4Experiment 33

Experiment 34

Experiment 36

Zr low purity

Calculation no reversion

Figure 4. Measured Wave Profiles at Different Impact StressesUnder high impact stresses (strong shock compression), the (particle velociy) wave

profiles for high-purity zirconium (red) and low-purity zirconium (green) display fea-

tures labeled 1–4, corresponding to the features described in Figure 3(d). In particu-

lar, the cusp at point 2 indicates the stress (or pressure) at which a solid–solid

phase transformation begins in the material, producing a second plastic wave P2.

Note that low-purity zirconium changes phase at a higher pressure than high-purity

zirconium, and its phase transformation is more sluggish—the pressure rises more

slowly as the phase transformation proceeds between points 2 and 3. The impact

stresses reached in Experiments 33 and 34 (also done on high-purity zirconium)

were lower than the pressures at which phase change begins, so the wave profiles

are smooth, indicating no phase change. These profiles were measured by Paulo

Rigg of Los Alamos.

Application. To implement a com-putational technique (say, a hydrody-namic calculation involving shockloading) that accounts for the behaviorof mixtures of phases, we must makesome additional assumptions. One-dimensional shock wave experimentsmeasure longitudinal stress, whereasthe thermodynamic properties dependupon pressure. For solids, longitudinalstress and pressure are different. Inmost cases, it is satisfactory to treatstrength effects as being elastic/per-fectly plastic, which is our usualassumption. We further assume thatcrystallites of the individual phases ina mixture are small enough that pres-sure and temperature are locally equil-ibrated, although the mixture need nototherwise be in thermodynamic equi-librium. We also assume that the shockresponse is rapid enough that allprocesses are adiabatic although notisentropic. Problem closure is achievedif rules are specified for the rates oftransformation between phases.Because our goal is to interpret experi-mental results, we use semiempiricalrules. In our most elementary models,we assume that the transformationrate between two phases is propor-tional to the calculated GFE differ-ence between the two phases andinversely proportional to a characteris-tic time for that particular transforma-tion. Our experiments show that char-acteristic times for a forward phasetransformation (from a low- to a high-pressure phase) are not always thesame as those for the reverse transfor-mation. We have shown this numericalapproach to be quite robust because iteasily handles very complex, nonequi-librium mixtures of many phases dur-ing computations of wave propagationin a phase-changing material.

Systems with complex geometriesoften defy direct measurement of thedetails of the response to impulsiveloading so that often the only alterna-tive is to develop computational mod-els. Many materials in systems of

interest undergo multiple, and some-times nonequilibrium phase changes,when they are shocked. The equationsof state and locations of phase bound-aries of these materials are measuredin simple experiments and the behav-ior is captured in a multiphase EOS.Then modeling provides the necessarybridge between the world of physicsand application.

Transmitted wave profiles that aremeasured in these experiments con-tain much more information on mate-rial behavior than just wave speedsand locations of phase boundaries.Most materials display a variety ofnonequilibrium effects that are promi-nent in shock and release experiments.Some phase changes are sluggish, andthis feature is reflected in the ratherbroad rise time in Figure 4. (See thebroad rise time between points 2 and3 compared with the sharp rise timebetween points 1 and 2.) The forwardtransformation sometimes slowsabruptly after only partial completion,clearly a nonequilibrium phenomenon,and this affects the speed of the wavebetween points 2 and 3 in Figure 4.There may be big differences in thespeeds of the forward and reversetransformations, which will affectboth the peak particle velocity and thedetailed structure of the release wave.Space limitations do not allow us togo into these aspects in any detail. Butit is important to recognize that theseexperiments provide unique data onphase change kinetics and nonequilib-rium effects that are invaluable forgeneration or validation of fundamen-tal theories of the phase changeprocesses at high strain rates.�

Acknowledgments

We gratefully acknowledge usefulsuggestions from John Vorthman,Carl Greeff, Paulo Rigg, and BillAnderson.



George T. “Rusty” Gray III received a Ph.D.in metallurgical engineering and materials sci-ence from Carnegie-Mellon University in1981.Rusty worked as apost-doctoral researcherand then staff member atthe Technical Universityof Hamburg-Harburg inGermany from 1982-1984. He is currently theleader of the DynamicMaterials PropertiesTesting and Modelingteam in the Materials Science and TechnologyDivision at Los Alamos National Laboratory.His research interests are focused on investiga-tions of the influence of material microstruc-ture on the dynamic and shock response ofmaterials.

Dennis Hayes received a Ph.D. in physicsfrom Washington State University in 1972. Hisresearch efforts at Sandia Labs focused onshock-induced phasechanges and radiationeffects. In 1996 hebecame the president ofLockheed Martin NevadaTechnologies where hewas responsible for execu-tion of all US DOE NVprograms. Dennis’ effortsat Los Alamos have focused on multi-phaseplutonium models, shock-melting in copper,twinning in uranium alloys and many otherendeavors aimed at extracting a physics under-standing of materials at the high temperaturesand pressures generated in a shock wave envi-ronment.

Robert Hixson received his bachelor’s degreein physics from California State University,Hayward in 1973 and his Ph.D. in physics fromWashington StateUniversity in 1980. After abrief postdoctoral appoint-ment, also at WashingtonState University, Robertjoined Los Alamos NationalLaboratory working in theApplied Physics group. Heremained here until 1982and moved to the Dynamic ExperimentationDivision where he is currently acting deputygroup leader. He has worked in the field ofshock compression physics for his entire pro-fessional career, authoring and co-authoringmany experimental research papers. Robert’scurrent research interests include studyingshock-induced phase transitions, leading todevelopment of multi-phase equations-of-state.

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