A Statistical Hot Spot

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A Statistical Hot SpotReactive Flow Model forShock Initiation andDetonation of Solid HighExplosivesA.L. Nichols, C.M. Tarver

This article w a s submitted to12'h International Detonation Symp osium , S a n Diego, California,August 11-16,2002

US . Department of Energy&:;Laboratory July 1,2002Approved for public release; further dissemination unlimited


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A STATISTICAL HOT SPOT REACTIVE FLOW MODEL FOR SHOCK INITIATIONAND DETONATION OF SOLID HTGHEXPLOSIVES*

Albert L. Nichols I11 and Craig M. TarverLawrence Livermore National Laboratory

Livermore, CA 94551A statistical hot spot reactive flow model for shock initiation and detonation of solid highexplosives developed in the ALE3D hydrodynamic computer code is presented. This model isintended to evolve into a physically correct description of the physical and chemicalmechanisms that control the onset of shock initiation via hotspot formation, the growth (01failure to grow) of these hotspots into the surrounding explosive particles, the rapid transition todetonation, and self-sustaining detonation. Mesoscale modeling of the shock compression andtemperature dependent chemical decomposition of individual explosive particles are currentlyyielding accurate predictions of hot spot formation and the subsequent growth (or failure) ofthese hotspot reactions in the surrounding grains. For two- and three-dimensional simulationsof larger scale explosive charges, a statistical hotspot model that averages over thousands ofindividual hotspot dimensions and temperatures and then allows exothermic chemical reactionsto grow (or fail to grow) due to thermal conduction is required. This paper outlines a firstapproach to constructing a probabilistic hot spot formulation based on the number density ofpotential ho'tspot sites. These hotspots can then either ignite or die out if they do not exceedcertain ignition criteria, which are based on physical properties of the explosive particles. Thegrowing hot spots spread at burn velocities given by experimentally determined deflagrationvelocity versus pressure relationships. The mathematics and assumptions involved informulating the model and practical examples of its usefulness are given.

INTRODUCTIONPhenomenological reactive flow models for the

shock initiation and detonation of solid highexplosives, such as the Ignition and Growth model'.*and the Johnson-Tang-Forest (JTF) model3 havebeen very successful in reproducing most of themain features of these reactive flows. The reactionrate expressions in these models depend upon theaverage compressions and pressures attained in thereacting explosive mixture rather than the local hotspot temperatures, which are known to control thereaction rates in the preferentially heated regions ofthe explosive charge. Thus there are somesituations, such as shock desensitization4, that arenot easily treated by these phenomenologicalmodels. Mesoscale modeling of the shockcompression5 and temperature dependent chemicaldecomposition6of individual explosive particles arecurrently yielding accurate predictions of hot spotformation mechanisms and the subsequent growth(or failure) of these hot spot reactions in thesurrounding grains. For two- and three-dimensionalsimulations of realistic size explosive charges, astatistical hot spot model that averages overthousands of individual hot spot dimensions and

temperatures and then allows the chemical reactionsto grow (or fail to grow) due to thermal conductionis required. Some simple statistical hot spot modelswere developed several years ago in one-dimensional hydrodynamic codes738,but practicalhot spot models in multidimensional codes are justbeginning to appear.

HMX and TATB used in this statistical model havebeen previously used to determine the critical hotspot temperatures and dimensions that will react andbegin to growg. The growth rates of these hot spotsdue only to thermal conduction were alsodetermined", and were shown to be too slow toaccount for measured shock initiation times and rundistances to detonation. Therefore physicalmechanisms, such as crack nucleation and growth,must be causing greatly increased reactive surfacearea as the pressure and temperature increase duringshock initiation and the transition to detonation.

In this paper, the formulation of a statistical hotspot creation model in the ALE3D hydrodynamiccode is presented. During shock compression, adistribution of hotspot sizes and temperaturesbased on the initial pore size distribution, explosive

The chemical kinetic decomposition models for


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particle size distribution, and the density of theexplosive charge is created. These hot spots thengrow to consume neighboring explosive particles orfail to grow as thermal conduction lowers theirtemperature before exothermic chemical reactioncan be completed. The growth rate of the survivinghot spots is then modeled by a statistical hotspotgrowth model normalized to experimental high-pressure deflagration rate data. The coalescence ofthese expanding hot spots as the temperature andpressure rapidly rise everywhere in the decomposingexplosive charge is also discussed. The extremelyrapid transition to detonation observedexperimentally is shown to be similar to a constantvolume explosion from a critical compression stateof the unreacted explosive to a state on the reactionproduct Hugoniot close to the Chapman-Jouguet (C-J) state. Examples of calculated shock initiation anddetonation wave propagation for an HMX-basedexplosive are presented. Examples of shockinitiation effects, such as desensitization andchanges in sensitivity due to initial temperature andparticle size distribution variations, that are difficultor impossible to compute with pressure andcompression dependent reactive flow models arealso presented.

STATISTICAL HOTSPOT MODELPROBABILISTIC HOTSPOT FORMULATION

The first phase in constructing the statistical hotspot model is the consideration of the distribution ofthose hot spots. First, consider the probability P, thatsingle hotspot of radius R will have reacted at agiven location in a volume V in the explosive. Thisprobability is given by:

4ZR3P,=- 3vIf there are NR of these hotspots randomly

distributed in space, then the probability that a givenlocation has not reacted P, is simply the product ofthe independent probabilities. Assuming that the hotspots are independently located, P, is defined as:

Taking the limit where the volume becomeslarge but the hotspot density remains fixed, Eq. (2)becomes:

(3)

Finally, the combined probability of a regionnot having been reacted P,&) is simply the productof the probabilities associated with each hotspotradius. Therefore the final expression for P, is:

(4)

The probability of not yet reacting is simply themass fraction of the reactant in a reactive flowformulation. The probabilistic formulation makes iteasier to consider a variety of different possibilities.For example, a similar line of reasoning can be usedfor two-dimensional hotspots (hotlines) and one-dimensional hotspots (hotplanes). The latter wouldbe useful for the modeling of shear banding as anignition source, for example. If all of such ignitionmechanisms could be defined, all that would berequired for a complete hot spot model is to multiplytheir probability functions together.HOTSPOT DENSITY MODEL

The process derived in the previous sectiondefines a mechanism for connecting the probabilitythat some quantity of explosive has been reacted tothe density distribution of hotspots. We define theprobability density of the hotspots as:


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t -Ida ldw p ,-- t

where ps(a,a) is the number density of hotspots thatignited at time a,and died at time a.The dirac-deltafunctions are used to define the size of the hotspotwith the assumption that the initial hotspot size is E,and that it then burns at a burn rate v out from thatinitial spot. The first term pA(R,t) represents thepopulation of hotspots of size R that are stillgrowing (active) at time t. The second term pD(R,t)represents the population of hotspots of size R thathave stopped growing (died) by time t. It isimportant to remember that even though a hotspotmay stop burning, the material that has burnedwithin that hotspot must still be counted as reacted.

Let us now define the following projections ofthe density function:

4 n -he)= - I d R R 3 p ( R y t )3 000

0t -

-- tca

The h term is just the negative of the log of theprobability defined in Eq. (4). The g and f terms arethe two- and one-dimensional active projections of

the density function, respectively. The number ofhotspots that are active at time t is A ( t ) ,and pB(t)is the number of hotspots created at time t. In thecurrent model, it is assumed that all hotspots activeat time t have the same rate of death p(t), that is:

-

With these projections, it is possible toconstruct a set of differential equations to couple thehigh order unreacted mass fraction with the muchsimpler active hotspot density.

dh-= 4 v (t E (t ) + 4 m 3 p , ($3at

IGNITION MODELIn order to complete the set of equations defined

in the previous section, we must define the rate atwhich hotspots are created. In order to model theexplosive process, it is necessary to choose anignition model that can encompass a variety ofphenomena associated with high explosives. Webegin by defining the initial density of potentialhotspots ppo. These potential hotspots can beanything from defects in the crystal lattice to voidsin the region between the explosive grains. For thecurrent model, we only limit ourseIves in that thepotential hotspot must transform into a roughlyspherical hotspot. Most postulated hotspot formationmechanisms involving void collapse predict that thespherical hotspots form upon full collapse''. Otherpotential hotspot formation mechanisms, such asshear-banding, would transform into roughly planarhotspots and thus are not considered in thistreatment. The shock process compresses these


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potential hotspots. If they are compressed to asufficiently high temperature, they will start to react(a hotspot). If the process is too weak, then thepotential hotspot will be destroyed without creatinga hotspot. Without such a process, any sufficientcompression of the explosive would lead to reaction,even that from an isostatic press. However, sinceexplosives do have strength, there must be sufficientforce to overcome that internal void strength beforeany changes to the potential hotspot density canoccur. The following phenomenological ignitionmodel captures these features.

@-e>AP*(P-P,)

Here K is the rate of potential hotspottransformation, and D is the constant death rate forpotential hotspots. PO is the ignition rate thresholdpressure that represents the internal resistance tovoid collapse. To prevent unrealistically largecollapse rates during numerical pressure spikes, P* isdefined as the saturation pressure. H is the heavyside step function, which is zero for all argumentsless than zero and one for everything else. Weoriginally envisioned a compression rate dependentignition rate, but such a rate can be extremely mesh-size dependent. More complex ignition models canbe formulated as this model evolves.PHYSICAL, INTERPRETATION HOTSPOTMODEL PARAMETERS

We have defined a total of 8 parameters, notcounting those associated with the equation of state,for this model. They are: PO,P*, A, p,v, D, ppo, andE. POis clearly related to the yield strength of theexplosive, and so we will use the yield strength inour model. The burn velocity v can beexperimentally determined by any of the standardburn rate measurement techniques, such as strand-burner and diamond anvil experiments. The valueof D should be chosen to match explosive shockrecovery experiments, so that the value of D is set

equal to the value of R for the shock pressure thatjust begins to ignite the explosive.A heuristic argument can be used to determine

ppo, and E. If one assumes that the initial hotspotvolume will equal the initial void volume and thatenough hotspots need to be created so that whenthey burn they will consume the entire explosive inthe reaction zone time T and initial void density py:

Another heuristic argument can be applied tothe ignition pre-factor A and P*. It is reasonable toassume that the rate of collapse of the void regions isproportional to the product of the void radius and theparticle velocity u in the shock wave. For relativelylow pressures, we can Taylor expand the volumechange using the adiabatic compressibility, and get:

where PO is the initial density, and c is the referencesound speed Comparing to the ignition definition inthe limit of small @-PO),we find

1

Although it might be tempting to use an ignitionform defined by the particle velocity, thecomplexities associated with making it work underthe variety initial conditions would be daunting. Forexample, changes in temperature would necessitate achange in the reference density within the root sign.Without such a change, hot systems could neverignite. We can handle the natural curvature thatcomes out of this formulation by an appropriatechoice of P*.


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EXAMPLE PROBLEMSSHOCK INITIATION

The phenomenological reactive flow modelIgnition and Growth' has been normalized to a greatdeal of shock initiation data. However, for differentinitial temperatures and particle size distributions, ithas to be reparameterized. The Statistical Hot Spotmodel has the necessary features to calculate suchinitial condition differences. The first exampleproblem is to drive a 2 cm wide by 3 cm long by 0.1cm deep block of explosive into a stone wall tocheck the two-dimensional shock initiationcharacteristics of the Statistical Hot Spot model. Aspreviously mentioned, an unreacted equation of stateis needed to describe the states attained during shockcompression. The Jones-Wilkins-Lee (JWL)equation of state is used with typical parameters foran HMX-based plastic bonded explosive.P =A(l-dRIV)e-R1v- B(lf~/R2V)e-~~+oE N (16)where P is pressure, V is relative volume, E is theinternal energy, o is the Gruneisen coefficient, andA, B, R1,and R2 are constants. For a typicalHMX-based plastic bonded explosive, the initial density is1.85 g/cm3, R1= 14.1, R2 = 1.41, o = 0.8938, A =9522 Mbar, and B = 0.05944 Mbar. This JWLequation fits the measured unreacted Hugoniot dataat low shock pressures and the von Neumann spikedata at high pressures12. The reaction products aredescribed by LEOS tables fit to product Hugoniotand isentrope states calculated by the CHEETAHchemical equilibrium code13. Mixtures of unreactedexplosive and reaction products are assumed to be inpressure equilibrium.

As discussed in the previous section, theignition and growth of reaction model has eightparameters: PO,P*, A, p,v, D, pp', and E. PO is theignition rate threshold pressure, which is related tothe effective yield strength at high strain rates and isset equal to 0.1 GPa for these calculations. P* is thesaturation pressure below which all hot spots areformed and is set equal to 10 GPa. The ignitionprefactor A is defined by Eq. (15) and, for HMX,isapproximately 2000 cm-pslg. The hot spot deathrate p is initially set equal to 5 ps-'. The constantdeath rate parameter for potential hot spot D is setequal to 11.3. The initial number of potential hotspot sites ppis assumed to be 1.4 x lolo ~ m - ~ .The

initial hot spot diameterE is assumed to be 1.5 x 1O4cm or 1.5 microns. Finally the reaction growth ratev is assumed to be a function or pressure asmeasured experimentally in strand burner^'^. Thispressure versus burn rate function is shown inTable 1.TABLE 1. PRESSURE VERSUS BURN RATEPRESSURE (GPa) BURN RATE (cm/us)1.0x 10" 1.016 x1.151 x 9.0683.842 x 1.763 x5.957 x8.077 x 6.988 x lo41.164 x 10" 1.016 x

3.599 x 1o-6

1.o 4.441 10-~Using these input values, the explosive was

driven into the stonewall at various velocities tostudy the buildup of the shock wave produced in theexplosive to a detonation. In agreement withnumerous experiments and previous models, thebuildup toward detonation occurred well behind theshock front, the transition to detonation was rapid,and the resulting detonation wave traveled at thecorrect velocity, 8.8 d p s . Table 2 lists the initialvelocities and the calculated times to completereaction. These reaction times are reasonable forHMX-based plastic bonded explosives. Figure 1shows the fraction reacted contours for HMXdecomposition at 2.33 ps for the 0.3 d p s initialvelocity case.

TABLE 2. INITIAL CONDITIONSAND TIMESTO COMPLETE REACTIONVELOCITY budus) REACTION TIMES (us)0.18 2.510.30 2.270.60 1.701.2 1.512.4 0.29


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FIGURE 1, Froctlo. Reacted Contours for HMX(Rad-DO reaetb; BIme-Cempkte remctiom) at233 pshr 88 i.itlalvelocity of0 3 d p 8The effects of changing various hot spot

pragqntcrs were investigatedusing an initial particlevelocity of 0.18 &pa or an initial shock pressureof approximbtely 1 GPa. Reducing the hot spotdtathratepammcterphm 5 to 1 reducedthetimeto conpletq d o n from 2.51 cup to 1.76 ps, andiDcrcaeiag p to 10 mused the fraction reaction torsm.dn below 0.45. Lowering the constant deathrate parameter D to 1.13 from 11.3 led to completed o n in 2 3 cup h s t d of 2.51 ps, and raising Dto 22.6 forccd~~rtacciOnto occurin9.3 ps.

One of the most important e&cts on shockinitiation is the initial particle size effect, whichoantcols both the initial n u m b of potential hot spotsites and the average size of these sites. Each solidmplosive exhibits a maximum shock sensitivity atcartain particle size. If the particles are too large,tbwe we fewm sites and thew sites arg easilycomp~&ssedmwltilrg in relatively low hot spottempmatumsP If the particles are small, there aremom dtes compressed to high temperatures, butmany of thesp hot spots lose their heat bysxmdW$mbfim thy cfplmwt and~ F OW PT*le 3shopvs the of changing the initial number ofpotential hot pp by an order of magnitude inw h direction. Large particle size formulations%wainfGWW si- ail4 tb!#l dfe Im elrock enSitiY6,TABLE3.E m C TOFBOT SPOT DENSITY8!i&dm& --1.4 x 1$ 0.0523 4.091.4 ~10'' 0.954 1.92

1.4 x 10" 0.976 0.114Table 4 shows the effbcts of iniM hot spotdkumtere 011 the restion times. Figure 2sbowa the

firaction ream4 forthe0.6-micron dimem oaet at 8time of 10.52 )IS when the shock wave has traveledthrough the 3 cm long charge. The 0.6 and 0.9micron results ignite approximately the correctamount of fiaction reacted for shock initiationstudies, and are reasonable hot spot sizes for malsolid explosive formulations. Since the d o n ratefalls rapidly with decreasing hot spot size andnumber of hot spot aites, the Statistical Hot Spotmodel is predicting the well-known particle sizeeffects on shock initiation of solid explosiveS.TABLE4. EFFECT OFHOTSPOT SIZE

onRe=ta l l2DdJ [email protected] lo4 0.979 0.291.5x lo4 0.954 1.!a9.0 x lo-' 0.1792 5.866.0 x lo-' 0.0322 10.52

The prcgsurc versus reaction rate frmction for hotspot growth shown in Table 1 is similar todeflagration rates observed in strand burnt= atprtssums below 1 GPa. For the higher pressumsreached during shook initiation and detonation, thegrowth rate fuaction was fit to recentdim& anvilcell (DAC)burn rate data on pure HMXu byRcaugh.6 These rates are shown in Table 5 ad aremuch ktcr than those in Table 1.

FIGURE 2. HM X Fraction Reacted Contours atl(M2pfor 0.6 pm M a m w Hog Spots impactedat an initial velocity of 0.18 mm/p


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T A B U 5. PPEssuRpr VERSUS DAC BURNRATES

1.OxlQ'3.Oso

5.0 x lo-s7.0 x lo49.0 x loa

c o w I D I N c ~ D E T o " w A v E sThe other main area where phenomenologicalreactive flow models sometimes have problems iswhen multiple shock or detonation waves collide,yielding regions of very high pressure and reactionrates. To test the ability of the StathtiicalHot Spotmodel to handle such problems, the explosive block.described above was initiatedon the top and bottomby imparting initial prcssum to start the d o nse


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7. Cochran, S. G. and Tarver, C. M., Shock Waves inCondensed Matter-1983, J. R. Asay, R. A. Graham andG. K. Straub, eds., Elsevier Publishers B. V., 1984, p.593.

8. Partom, Y., Seventh Symposium (International)onDetonation, Naval Surface Weapons Center NSWC M P82-334, Annapolis, MD, 1981, p. 506.J. Phys. Chem. 100.5794 (1996).

International Detonation Symposium, Office of NavalResearch ONR 33300-5, Snowmass, CO, 1998, p. 599.

Eleventh International Detonation Symposium, Officeof Naval Research ONR 33300-5, Snowmass, CO,1998, p. 768.

Green, L. G., Propellants, Explosives, Pyrotechnics,l&117 (1993).

9. Tarver, C. M., Chidester, S. K., andNichols, A. L. III,10. Tarver, C. M. andNichols, A. L. HI,Eleventh

11.Conley, P. A., Benson, D. J., and Howe, P. M.,

12. Tarver, C. M., Urtiew, P. A., Chidester, S.K., and

13. Fried, L., Howard, W. M., and Souers, P. C., EXP6:A New Equation of State Library for High PressureThermochemistry, paper presented at this Symposium.

International Detonation Symposium, Office of NavalResearch ONR 33300-5, Snowmass, CO, 1998, p. 872.

14. Maienschein, J. L. and Chandler, J. B., Eleventh

15. Farber, D. L., Bug, J. M., andRuddle, C., ShockCompression of Condensed Matter-2001, Furnish, M.D., Thadhani, N. N., and Horie, Y, eds. CP-620, A PPress, New York, (2002).

16. Tarver, C. M., Cook, T. M., Urtiew, P. A., and Tao,W. C., Tenth International Detonation Symposium,Office ofNaval Research ONR33395-12, Boston, MA,1993,p. 696.

17. Whitworth, N. J. and Maw, J. R., Shock Compressionof Condensed Matter-1995, Schmidt, S. C. and Tao, W.C., eds., AIP Conference Proceedings 370, AIP Press,New York, 1996, p. 425.

A Statistical Hot Spot

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