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A MACROSCOPIC APPROACH TO THE INITIATION AND DETONATION OF
CONDENSED-PHASE ENERGETIC MIXTURES
J. Nunziato, M. Baer
To cite this version:
J. Nunziato, M. Baer. A MACROSCOPIC APPROACH TO THE INITIATION AND DETONATION OF CONDENSED-PHASE ENERGETIC MIXTURES. Journal de Physique Colloques, 1987, 48 (C4), pp.C4-67-C4-83. �10.1051/jphyscol:1987403�. �jpa-00226635�
JOURNAL DE PHYSIQUE
Colloque C4, suppl6ment au n09, Tome 48, septembre 1987
A MACROSCOPIC APPROACH TO THE INITIATION AND DETONATION OF CONDENSED-PHASE ENERGETIC MIXTURES
J.W. NUNZIATO and M.R. BAER
PZzstd and Thermal Sciences Department, Sandia National
Laboratortes, Albuquerque, NM 87185, U.S.A.
Rbsum6.- I1 est generalement admis que pour des conditions de chargement donnees, l e s techniques de microstructure e t l a reaction chimique jouent un r61e majeur dans l ' i n i t i a t i o n des explosifs en phase condensbe granulaire e t l a t r a n s i t i o n v e r s l a detonation qui s ' e n s u i t . Dans un e f f o r t de comprehension des mecanismes fondamentaux, notre recherche theorique et experimentale s ' e s t f o c a l i s e e s u r l16valuation p r e c i s e de l a microstructure e t l e dbveloppement des modeles mixtes phase multiple continue pour d e c r i r e les e f f e t s observes de l a t a i l l e de l a p a r t i c u l e e t de l a porosite.
La formation des p o i n t s chauds e t l e s mecanismes de r e a c t i o n chimique.
comprenant l a decomposition thermique des p o i n t s chauds et l a combustion du g r a i n controlee par s a surface sont au c e n t r e de l t i n t 6 r 8 t . Dans c e t a r t i c l e , nous donnons une revue de n o t r e recherche s u r c e s s u j e t s et discuterons nos conclusions dans l e contexte d ' a p p l i c a t i o n s specifiques aux e x p l o s i f s granulaires, HMX e t HNS. Un accent p a r t i c u l i e r est m i s s u r 1 ' B t a t a c t u e l d e n o t r e comprehension e t l e besoin de f o c a l i s e r plus clairement n o t r e a t t e n t i o n s u r une etude fondamentale de l a chimie impliquee dans les hautes pressions.
A b s t r a c t . -- It is generally accepted that for given loading conditions, microstructural processes and reaction chemistry play a major role in the initiation of condensed-phase granular explosives and the subsequent trarisition to detonation. In an effort to un- derstand the fundamental mechanisms, our theoretical and experimental research has focused on quantifying the microstructure and developing continuum multiphase mix- ture models to describe the observed effects of particle size and porosity. Of central importance is the formation of hot spots and the chemical reaction mechanisms, includ- ing the thermal decomposition of hot spots and surface-controlled grain burning. In this paper, we will review our research in these areas and discuss our conclusions in the context of specific applications to the granular explosives, HMX and HNS . Particular ernphasis is given to the state of our current understanding and the need to focus more clearly on the fundamental chemistry involved at high pressures.
1 I n t r o d u c t i o n
The processes associated with the initiation and growth to detonation in porous, granular explosivc~s arcb krtown to be directly related to the loading conditions and the details of the microstructure of the mixture. In the case of impact loading, the essential factor in wave growth is the formation of local regions of elevated thermal energy, i.e., hot spots, having temperatures higher than the average temperature expected from shock compression. When sufficient thermal energy is generated locally, ignition occurs and the subsequent chemical energy release results in the developnient of a strong pressure disturbance behind the wave front which amplifies as it
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987403
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overtakes the front, ultimal,ely resulting in detonation. Since the concept of hot spot initiation was first proposed by BOWURN & YOFFE [ I ] , a number of possible physical mechanisms for hot spot formation have been proposed 12-41, The common aspect of these proposals is the fact that initiation depends subtly on the microstructural features of fhe material, i.e., pore size, grain surface area, etc. The dependence of the observed phenomena on these features often results in confusion and disagreement as to the precise mechanism operable in a specific instance. Nevertheless, the concept of hot spots as a necessary ingredient in any theory of shock initiation appears to be firmly established.
The major issues in describing the initiation and growth-to-detonation of condensed-phase granular explosives concern: (1) the identification of the local thermal and mechanical processes (e.g., hot spot formation, high temperature gas flow in the connected porosity, etc.) which are most important for the loading conditions of interest, (2) the development of the micromechani- cal models which describe those processes, (3) the characterization and analytical description of the important reaction mechanisms, and (4) the incorporation of these models in a continuum theory suitable for implementation in numerical simulation codes. In this paper, we discuss these issues in the broadest sense for granular explosives without binders. In Section 2, we focus principally on the role of the microstructure in the formation of hot spots and review the chemical reaction models which have been proposed to describe, in a global sense, the ignition and combustion of hot spots and their effect on wave growth 15-13]. The concepts here can be confusing and misleading as a result of the strong desire to reduce the explosive decompo- sition process to its simplest terms. As a result, the existing models for hot spot initiation are largely empirical. Here we take what we view as a more appropriate approach and consider granular explosives as two-phase mixtures (the continuum theory of mixtures is outlined in an Appendix) with two reaction mechanisms operable; one reaction mechanism representing the thermal decomposition of the hot spots and the other characterizing the surface-controlled grain burning. In this model, the hot spots nucleate in the distorted regions of the individuar grains during pore collapse and the decomposition rate is determined by the hot spot temperature and the volume of heated material. Subsequent combustion is controlled by the surface area of the grains and the gas pressure. The initiation of grain burning occurs when the surface of the grain exceeds the melt temperature. While this model is not based on detailed kinetics, it does capture the essence of the combustion process to a reasonable degree. To illustrate the utility of this phenomenological approach, we discuss the low velocity impact of HMX in Section 3 and shock initiation of HNS in Section 4.
2 H o t S p o t s a n d Phenomenological Chemical Reaction Models
The initiation and growth to detonation of condensed phase granular explosives has been under investigation for many years, and the importance of discrete hot spots in the process is now well recognized. However, little progress has been made in determining the precise physical mechanisms which control the nucleation, growth, and coalescence of hot spots, and the corresponding chemical decomposition of a given explosive for a specified loading condition.
The processes involved are extremely complex. Consequently, there remains a cert,ain amount of ambiguity in the interpretation of data and non-uniqueness in the theortetical models which have been proposed.
In an attempt to clarify the picture, we focus our attention on granular explosives without binders. In this context, two principal mechanisms for hot spot formation have been identified as physically realistic. One mechanism is associated with the interaction of the pressure wave with the density discontinuities within the explosive [2] and the collapse of the available pore space (Fig. 1). Localized heating results from the elastic-plastic work done in a thin layer of material bounding the pore space and jetting across the space during the loading process [4,8]. Viscous
heating in narrow regions of high shear is another possible mechanism for the nucleation of hot spots ;3]. These regions may either occur in the neighborhood of grain boundaries where large frictional forces may exist or occur within individual grains, similar t o shear bands in metals.
Regardless of which mechanism is operable, it should be readily apparent t h a t the number density of hot spots formed in the explosive, their temperature and their volume is directly related t o t h e loading rate, the amplitude, and the duration of the pressure wave, as well as the number density of inhomogeneities in t h e material. Another important factor controlling the strength of t h e hot spots is thermal diffusion. For time scales comparable t o wave transit times, diffusion length scales can be of the same order as the hot s p o t volume and therefore local heat transfer can significantly reduce the hot spot temperature.
Once ignition occurs a t localized hot spots, the remainingchemical energy within the material is released and subsequent.ly coupled into the wave motion. This process is believed t o occur in a two-step fashion. In t h e first step, chemical energy is released in t h e region behind the shock front due t o rapid thermal decomposition of the hot spots. Although the total energy release in t h e process is relatively small, t h e hot spots d o provide sites of high pressure and high temperature gas essential t o the initiation of subsequent combustion. T h e second step concerns a heat transfer dominated, grain burning mechanism which controls the growth t o detonation. Once again, t h e microstructural features of t h e explosive, such as specific surface area, play a major role in how the combustion proceeds. Here large pressure disturbances are generated o n a longer time scale which overtake the wave and progressively amplify the wave until detonation conditions a r e reached. However, in order t o achieve sufficient rates of chemical energy release, computational results have shown t h a t t h e grain burning mechanism must be modified - for example, by including a higher order pressure dependence. Physically this may be due t o several factors; t h e increasing number of hot spots which are generated as a result of strengthening t h e initial wave amplitude, the ignited surface area being greatly increased as a result of grain fracture, or by enhanced gas phase combustion.
Several previous st,udics h a v ~ attempted t o develop a macroscopic description of initiation and growth t o detonation in energetic mixtures bascd on these physical ideas. However, fcw have addressed the effects of t h e microstructure explicitly in order t o define a hot spot temperature anda, complete description of the reaction chemistry is lacking. Rather, t h e reacting material is often treated as a mixture of two coexisting phases consisting of the unburned condensed phase explosive and the gaseous reaction products. A simple mixture law in terms of a single reaction coordinate is used to express t h e specific volume and energy of t h e mixture as weighted sums of these variables for each phase. Each phase is also assigned a suitable equation of state and both pressure and temperature equilibrium are assumed. T h e decomposition of the hot spots, as well as the subsequent reaction of the explosive, are then described by one or more evolutionary equation related t o the reaction coordinate. For example, COCHRAN 151 developed a statistical treatment of hot spot nucleation and growth. In this case, the reaction coordinate is given in terms of the number density of hot spots and their volume, both of which are governed by evolutionary equations t h a t account for the effects of thermal diffusion and grain burning l'rorn spherical sites. 111 l.he sarrtt, spirit, I,EE and TAR.VER [ti; l~avc? considered a sir~glc kirlctic equation for the chemical reaction rate containing both "ignition" and "growthn terms. The
"ignition" term contains a factor t o simulate hot spot formation as a result of plastic work a t grain boundaries. This results in the burning of a small fraction of the explosive during the passage of t h e shock front. T h e "growth" term is related t o the mixture pressure raised t o a power greater than one. JOHNSON, et al. 171 have extended the "ignition and growth" model by considering two reaction c0ordinat.e~; one for the total extent of reaction and one for the thermal decomposition of the hot spots. The resulting kinetic equation for the total extent of reaction makes explicit use of the known thermal deconlposition kinetics for the explosive and
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generalizes the "growth" term t o match laminar grain burning at. low pressures and the P o p plot d a t a a t high pressures. This approach significantly improvt=s the model's ability t o predict the rapid roll-over t o detonation observed in experiments. In general, these phenomenological models (also known a s single temperature, hot spot models) do a reasonable job of reproducing the experimental d a t a o n sustained shocks in HMX-based PBX9404 to which they have been calibrated.
BAER, KIPP, HAYES, and NUNZlATO 18-13] have taken a somewhat different approach which is based on t h e theory of multiphase mixtures. These investigations have focused on quantifying the microstructural processes which lead t o t h e formation of the hot spots and on explicit calculations of the hot spot volume and temperature. In the early work 18-11], the irreversible work done during loading due to the collapse of the available pore space ( i . e . , compaction) was assumed t o be dissipated in a specified hot spot volume, such as the initial pore volume. While this formulation gives reasonable hot spot temperatures a t low pressures, the models were unable t o adequately predict the different hot spot volumes t h a t would result from different loading rates, different wave amplitudes, and thermal diffusion. K I P P 1121 and BAER and NUNZIATO [13] have overcome this difficulty by formulating an evolutionary equation for the hot spot volume:
and a n appropriate equation for local energy balance t o compute t h e hot spot temperature T f .
K I P P 1121 based t h e form for t h e function F on shear band concepts similar t o those developed for metals, a n d again matched t h e sustained shock wave d a t a on PBX9404.
Here, we follow t h e multiphase mixture approach proposed by BAER and NUNZIATO [13]
and ascribe the hot spots t o the distorted grain volume t h a t results from compaction. In this description, the volume of material assigned t o the hot spots due t o compaction is defined as the increase in solid volume fraction above the initial state, 4::
and can be computed directly from t h e evolutionary equation (A14). To formulate a n equation for local energy balance for the hot spots, we consider t h e work
due t o t h e configurational changes of the dispersed solid phase induced by the solid phase pressure, p,. It is argued that, during pressure loading, this energy is partitioned t o hot-spo1,s prior t o t h e emergence of chemical energy release. Thus, this energy establishes localized heating which, by t h e local processes of heat transfer and combustion, ultimately transfers to the gas phase. Specifically, the increase of the internal energy in t h e hot spots equals the compact,ion work less t h e energy dissipated by conduction:
where C, is the solid phase thermal capacity, h is the thermal exchange coefficient, SH is the surface area of .the hot-spots over which energy loss occurs, N H . is the number density of hot.
spots, 7; is the bulk solid temperature, 7, is the solid phase density, and c z in the mass exchange.
As a representative geometry, spherical hot-spots are assumed t o originate a t contact points in the granular material. During compaction, this displaced material fills t h e void spaces of the granular material and the hot-spot diameter is scaled t o the pore diameter d,,:
Then N H and SH are given by:
N H T= ~6 ( d , ~ - 4;) , SH = ~ d , i ( 6 )
and, from classical conduction theory (Nusselt), the exchange coefficient h for a spherical geom- etry is given by:
It is of interest to note that given a mechanical loading with a time variation of
p , - p , - t r n ( m > O ) , (8)
it can be shown from ( A 1 4 ) and ( 4 ) that t h e early time solid volume fraction and hot-spot temperature variations a r e
g5s - 4;
-
t r n f l and T: - T," - t m (9)which implies t h a t the hot spot temperature rise follows directly with t h e rise rate of t h e solid phase pressure.
Once t h e hot spots a r e formed, t h e combustion process is assumed t o proceed in two steps;
rapid thermal decomposition of t h e hot spots followed by grain burning. In terms of multiphase mixtures, the corresponding reaction rate model is a n expression for t h e mass exchange cb,
where t.hr shape fiinction 0 2 reRect.s tht. spherical divergence of the burn front, d , is the surfa.ce Irloari grain diameter (cx d , , ) , p, @,y, i ( I & , ) p , is the rnixturc pressure, Z is the frequency fact.or, and T ' is t h e act,ivation t,emperaturc. The correction factor a l , applied t o the Arrhenius kinetics, is included t o account for possible discrepancies a t high pressures and temperatures. As will become evident, there are real questions a s t o t h e validity of low pressure, time-to-explosion d a t a in this regime, particularly in view of the fact t h a t solid phase reactions may play an important role in the thermal decomposition process. ROGERS 1141 points out t h a t the solid phase reactions in time-to-explosion experiments are extremely difficult t o measure, and the reaction rates may vary widely with changes in crystal purity and perfection. The grain burning port,ion of the reaction rate model is also open to some question. It, is now believed t h a t the grain burning, being surface controlled condensed phase and gas phase combustion processes, cannot begin until the surface of the grains approach the melt temperature of the explosive.
Moreover, the pressure dependence of the burning is generally unknown. While many previous studies have used n as a fit,ting paramctter t o calibrate their model, we have chosen t o assume a lincwr deprrrdence on t h e gas p r e s s ~ ~ r e characteristic of la.minar burning (n = I ) . This choice is motivated by the fact t h a t there appears t o be a greater uncertainty in the parameters governing t h e thermal decomposition reaction a t high pressures.
3 L o w Velocity Impact o f t h e Granular E x p l o s i v e HMX
During the last decade, there has been considerable interest in the phenomena of deflagration-- to--detonation transition (DDT) in granular explosives. This process begins with the ignition of a few grains by some external energy source and proceeds slowly a t first since it is heat conduc- tion dominat,ed. The high tempera.ture product gases generated during this decomposition can
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penetrate t h e unreacted material via t h e connected porosity a r ~ d , by prehrat,ing the grains, aug- ment the combustion such that a flame is driven into t h e explosive a n order of magnitude faster than by conduction alone. The existence of this convectively-driven flame front was confirmed experimentally by GRIFFITHS & GROOCOCK 1151 in which they measured the run distance to detonation in a column of HMX. They showed t h a t t h e onset of detonation was well removed from the location a t which combustion began. Subsequent experiments by PRICE & BER- NECKER 1161 and CAMPBELL [17] have demonstrated the importance of grain compaction in the flame acceleration process and load transfer t o t h e solid grain structure in columns under strong confinement. Physically, these effects result from the significant gas pressures which are attained due t o t h e chemical reactions and t h e choked flow in the connected porosity ahead of t h e flame front. T h e local drag forces and the high gas pressures act like a piston on the solid grain structure and t h e disparity in the gas and solid pressures results in the compaction of t h e porous material ahead of t h e flame. This reduces the permeability leading to further increases in t h e gas pressure. This process continues and is a key factor in the formation of a low amplitude shock wave which characterizes the compressive reaction phase of DDT. Once the wave is formed, hot spots again play a major role in t h e growth of t h e wave t o detonation.
BAER & NUNZIATO [18] have used the two-phase mixture model outlined in t h e Appendix t o describe D D T in one-dimensional columns of HMX. T h e model neglected t h e presence of hot spots and assumed t h a t the decomposition of t h e granular explosive occured a s a result of laminar grain burning; t h a t is, t h e mass exchange (2) was written as
subject t o the ignition criterion t h a t t h e reaction begins when t h e mixture temperature T, =
4..Tq + (1 - +,)T, exceeds the melt temperature of solid HMX. At a reference pressure of p* = 0.1 GPa, t h e burn velocity V, of HMX is approximately 0.1 m / s (BOGGS [19]). Note t h a t , consistent with dai.a., srr~all ~)art.icles and particles less dc~nscly packed tcwd t.o burn faster.
'Yo cornplete the model, specific forms for the momenturrr and energy exchanges were de- veloped consistent with experimental d a t a and thermodynamic arguments. Compaction in the explosive was assumed t o be governed by the evolutionary equation (A14) and t h e dependence of the configuration pressure p, on the solid volume fraction was evaluated from quasi-static loading data. These models, along with the appropriate equations of state, were incorporated into a one-dimensional Eulerian code solved using t h e Method of Lines. Numerical calculations were made of D D T in a 100 mm column of HMX packed t o 70% initial density with uniform particles. T h e computations used uniform zoning and assumed t h a t a few initial zones of the column were ignited by low pressure gas a t the adiabatic flame temperature. Thus, the calcula- tions focus on the regimes of convective c o n ~ b ~ l s t i o n , compressive reaction, and detonaf.ion. The predicted burn front trajectory, including the burn velocity, the detonation velocity, and the run distance t o detonation, agreed well with the experimental d a t a for 100 p m HMX obtained by PRICE: & BERNECKER 1161 using ioniza.tion pins embedded in the granular explosive. To furl.11rr tc,sl, t h e t.heory, c:alr~~lal.ions wvrr also done for variol~s grain sizes and t h e predicted run distance was compared t o data [l6,17]. For d, > 40pm, the theory predicts t h e correct trends in t h a t the run distance t o detonation increases with increasing grain size. However, the theory consistently predicts longer run distances than observed and, for grain diameters less than 40 pm, t h e trend is exactly opposite t o what is observed in the experiments. In evaluating these results in t h e context of previous work on shock initiation, it was concluded t h a t hot spot for- mation plays a significant role in the compressive reaction regime of DDT and t h a t this effect is most important for fine particle HMX.
To investigate hot spot formation in the low velocity regimes encountered in DDT, SAN-
DUSKY & BERNECKER 1201 have carriea out a series of pist.on impact experiments in which columns of 73% dense, Class D (870 p m ) and 44% dense, Class E (15 pm) HMX were sub- jected t o long duration (> 200ps), low level shock loading (< 0.18 GPa). The columns were packed in tubes of varying strength to a uniform density over their entire length of
-
140 mm.The primary instrumentation was radiography of metallic tracers embedded in the core of the porous column. D a t a was obtained for impact velocities ranging from 0.074 t o 0.308 mm/ps and, in each experiment, the impact resulted in the propagation of a compaction wave into the explosive. There was clear evidence that reaction commenced with the generation of hot spots during the compaction process; however, there was a delay time r, between impact of the column and the detection of reaction which varied inversely proportional with the square of t h e mixture pressure. After detection of reaction, the growth observed varied significantly depending on t h e impact velocity, the confinement and the required run distance, which in these experiments could be quite long. To illustrate the different phenomena observed, Fig. 2 shows the piston and burn front trajectories for impact velocities of 0.125 and 0.267 m m / @ for 73%
dense Class D HMX confined in a lexan tube. For the 0.125 m m / p s impact condition, a weakly luminous reaction front is generated by hot spots behind t h e compaction front. This reaction front overtook t h e compaction wave t o produce a steady low velocity combustion wave which propagated t h e remaining length of t h e column. For the higher velocity impact, the compaction front itself is luminous, indicating t h e presence of hot spot generated reaction very near the front. In this case, the wave accelerates rapidly and detonation is achieved in approximately 50 mm. For the fine grain, Class E HMX, considerably higher impact velocities were required t o initiate reaction; however, it appears t h a t once ignition occurs, t h e rection proceeds more rapidly.. This would be expected on t h e basis t h a t for fine grain material there would be a higher number density of hot spots with lower energy for the same impact condition. However, once ignition occurs, t h e higher surface t o volume ratio would result in significantly faster reaction rates. VON HOLLE 1211 observed this same phenomena using infrared radiometry t o directly examine hot spot. formation and rea.cl.ion growth on the back surface of shocked HMX samples.
In a n cKort t o dcscribc Illcso obsc!rvatioms alld account for hot spot for~rlation in DDT, wc have extended t h e muitiphase mixture theory in several important respects. In particular, we have adopted the hot spot formation-and global reaction model outlined in Section 2 with the shape function a2 given by
This expression represents well t h e surface t o volume ratio of both simple cubic and face-centered cubic spherical grain structures during compaction The grain burning was assumed t o have linear pressure dependence (n = I ) and initiate only when t h e surface temperature exceeded the melt t.emperature. In the presenl theory, the surface temperature was calculated a t each point in the mixture based on an approximate analysis which equates the rate of energy increase due t o the inc rcase in t h e bulk grain temperature to the energy the particle receives by the convective pd5 f l o ~ j)dsl t h r particlr. The rapid thcrrndl dctornposition of t h e hot spots was assumed t o follow the Arrhenius kinetics for solid IIMX given by RODGERS 1141.
This theoretical model was incorporated into the one-dimensional code described previously and calculations were carried out for the 0.125 m m / p s impact condition (in the code, the bound- ary condition is one of the particle veiocit,y and consequently, we used the value of 0.100 mmfps as given by experimental observation). The calculation yielded t h e correct compaction velocity of 0.45 mm/ps; however, no combustion front was observed t o form a s a result of hot spot nucleation and growth. Specifically, hot spots did form, but they were not of sufficient strength t o melt t h e surface layers of t h c grains and initiate the grain burning process. Increasing the
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frequency factor by a fact& of four produced increased compaction wave speed a s supported by the hot spot combusion; however, grain burning was not initiated. We interpret t,his as clear evi- dence t h a t the thermal decomposition kinetics based on time-to--explosion data obtained a t low pressures are simply not applicable t o high pressure loading conditions. Rather, more detailed analyses of the reaction mechanisms operable are required, along with additional experimental d a t a on t h e actual chemical species generated in the hot spots. Laser spectroscopic methods may offer one way t o determine the desired information. T h e crucial issue, however, will be in devising t h e right experiment so t h a t it yields the correct data.
With inadequate d a t a on t h e thermal decomposition of the hot spots, we decided t o replace that portion of the rate expression with a term relating the experimentally observed delay time
7, t o the mixture pressure p,:
This is certainly a phenomenological approach; however, it does permit u s t o evaluate t h e other aspects of t h e model and its ability t o predict the observed behavior. The constant of proportionality b was determined by obtaining t h e best fit t o t h e delay time observed for 0.125 mm/ps impact condition. The calculated results agree fairly well with the overall wave behavior observed (Fig. 2). However, unlike t h e data, t h e calculations predict t h a t the hot spot reactions occur near the front and t h a t detonation is achieved. The release of chemical energy near t h e compaction front due t o t h e hot spots again points t o the need for better descriptions of the dominate reaction mechanisms; the model (13) is oversimplified a n d does not properly capture t h e ignition delays a t lower pressures necessary t o model the generation of the weak combustion wave which develops behind the front. The fact t h a t the model predicted detonation should not be surprising; one dimensional calculations assume infinite confinement and thus will always predict t h e growth t o detohation if sufficient energy is available t o initiate t h e grain burning.
T h e two-din>ensional effects due t o weak confinement offrrs a probable reason why a steady, low ve!ocit.y combustion wave was c:xperirnentally observed in this case. IJsing, the same parameters, we also calculated the wave behavior for the 0.267 mm/ps irnpact condition (Fig. 3). There is quite good agreement with the experimental d a t a in this case. Initially, there is a low velocity compaction wave and, after a delay time similar t o t h a t observed, combustion occurs very near the front which enhances the compaction'process. This leads t o a higher compaction velocity in agreement with experiment which builds to detonation a t the front in about 40 mm. Clearly, the simplified model (13) works quite well here.
Similar calculations were made for one other impact condition for Class D HMX and for two impact conditions for Class E HMX using t h e same model parameters. It is of interest to note t h a t for t h e fine particle Class E material, a higher impact velocity was required t o initiate detonation as in the experiments and the calculated initiation did begin a t some distance behind the compaction front. The run distance t o detonation calculated for Class D material for various impact velocities is shown in Fig. 4 along with the experimental d a t a of SANDUSKY RL HRR.NI':<:KER 1201. In general, the model agrees well witah the observat.ions. It is parl.icularly inf.eresting that t h e model predictions would appear t o extrapolate nicely to fit, the experimental d a t a of DICK 122) obtained in wedge tests for 65% dense, Class A (100 pm) HMX.
4 Shock Initiation of the Granular Explosive HNS
Experimental and theortical studies by SIJEF'FIELD, el al. 1231, HAYES & MITCHELL 1241, HAYES 191, HAYES et al. [lo], and SETCHELL & TAYLOR 1251 of the shock initiation of granular hexanitrostilbene (HNS) have helped t o substantiate our view of the microstructural processes which control hot spot formation arld reaction growth. However, in interpreting t h e
t~sperimental observations in tJle context. of the react ion mechanisms operable, there appear t o be some peculiarities special to this cxplosive which warrant some discussion.
To investigate t h e shock initiation of HNS, plate impact experiments were carried out for two different particle sizes; HNS--I (21 q m ) and HNS-I1 (37 pm) 123,241. These experiments utilized a light gas gun to cause an IINS/fi~sed silica projectile t o impact a fused silica target for velocities in t h e range of 0.5 t o 1.3 mrn/ps (pressures of 1.25 - 5.28 GPa). Using velocity interferometry, particle velocity histories were obtained in the fused silica target and, because of the loading configuration, these histories are in direct one-to-one relationship with t h e pressure a t t h e HNS/fused silica interface. The d a t a for HNS-1, shown in Fig. 5, is characterized by a square pulse reflecting the wave transit through the fused silica, a n unloading t o a point on the HNS Hugoniot, and then a pronounced delay after the HNS is shocked, which is identified with the formation and thermal decomposition of the hot spots. This delay is followed by a pressure excursion which is produced by the grain burning and ends at t h e pressure believed t o correspond t o reaction completion. The delay time is a decreasing function of increasing impact pressure and the subsequent reaction becomes more rapid for increasing impact pressure. Thus, t h e d a t a gives a direct means for evaluating t h e equation of state of t h e reactant, in addition t o valuable insight into the character of the shock induced decomposition kinetics.
HAYES (91 interpreted these observations in terms of a quantitative two-temperature model which assumes t h a t the hot spots are formed a t pore sites as a result of the irreversible work done by pore collapse during the shock loading. The Hugoniot energy is partitioned such t h a t t h e bulk material is isentropically compressed and t h e balance of t h e energy is dissipated in the hot spot (pore) volume. This approach has been substantiated by calculations of the interaction of a shock with a single pore (Fig. 1) and provides t h e initial conditions for evaluating the decomposition kinetics for the measured pressure histories. To complete the analysis, HAYES 191 assumed t h a t t h e pressure rise during the decomposition was directly related t o t h e solid mass fraction ( e , = p,/p) reacted and that, for mass fractions less than t h a t in the hot spots, the decomposition was 1,herrnally coritrollrd. Reaction completion was then achieved by burn front,s whic11 propagal(, radially o t ~ t . Fro111 c\ach hot spot, at. a velocity 1/ 1.0 bc inferred from t,he bulk pressure excursion corresponding to mass fractions greater than that in t.he hot spots.
Analysis of the d a t a in the context of this model leads t o several major observations. (1) The calculated delay times corresponding t o hot spot decomposition are shorter than expected based on extrapolated low pressure thermally -activated kinetics. This again supports the notion t h a t the Arrhenius parameters determined from time-to-explosion experiments are suspect a t high shock pressure. ( 2 ) Contrary to experimental d a t a on other explosives, the burn velocity does not have a st,rong dependence on the gas pressure but does depend on t h e initial shock pressure pi above a threshold of 2 (;Pa:
where p*: = 1.0 GPa. The unique kinetic mechanisms which lead t,o this phenomenon are unknown. (3) T h e hot, spot, l,en~pt,ra.t.urrs arid the burn velocities a.re similar i l l HNS samples 11a.villg diKcre111, initial g r . a . i ~ ~ sissc,s if 1.l1c. c~xplosivcxs arc3 s l ~ o c k ~ d lo 1.11(. sarllc' I)rtrssurc. Thr fact t h a t the hot spot, t.emperalure is independent of grain size is cor~sistent with the assumption t h a t the hot spots form instantaneo~lsly in the shock front and not over some characteristic length/time scale. T h e evolutionary equaf.ion for the pore collapse (A14) introduces this type of concept in order t o account for t,he effect, of loading history on hot, spot formation. Finally, even with the burn velocity independent of particle size, HNS-I, having a larger surface t o volume ratio, still exhibits a faster overall rate of chemical energy release than the larger grained HNS-I1 a t t h e same pressure. This conclusion is consistent with previous experimental observations on other explosives in t h a t the senait.ivity t o shock impart, increases with decreasing particle size.
C4-76 JOURNAL DE PHYSIQUE
Howevel, this is only true up t o d ~ O ~ T I L for H \ > . SETCHE1,L & TAYLOR 1251 have noted that this 1 rend reverses for sufficiently smali partic!es.
T h e kinetic information developed by HAYES (91 have been used a s a basis for the rate expressions used by HAYES et a1 [lo] in d multiphase mixture model based on the theory outlined in the Appendix. The mass exchange is again represented by two terms; the first for thermal decomposition of the hot spots and the second for the grain burning:
T h e geometric shape functions Al and Az are very similar in their effect to those used prevoiusly, differing only in detail. Note that, since Arrhenius type kinetics do not appear t o be valid a t these high pressures, the delay time calculated for the hot spot decomposition 191 was fit as as function of t h e shock pressure. This theory has been incorporated into a one-dimensional Lagrangian wave propagation code, assuming t h a t each phase moves with the particle velocity of the mixture. T h e code was used t o simulate the plate impact experiments described previously and the calculated particle velocity histories are also shown in Fig. 5. Comparison with the d a t a is reasonably good, indicating t h a t the kinetic information deduced from the d a t a is self- consistent. However, it is important t o emphasize t h a t it may not be unique. Altering the assumptions used t o reduce the d a t a could have lead t o a somewhat different model which could also be self-consistent. This uncertainty in the reaction kinetics for HNS has lead RENLUND
& TROTT [26] t o explore laser spectroscopic methods t o gain further information on the CN radicals which evolve during detonation of unconfined samples. In addition, SETCHELL 1271 has been developing time-resolved emission spectroscopy t o investigate t h e occurrence of CH, CZ, and C N radicals during shock initiation of confined HNS samples a t 3.0-4.0 GPa.
5 Conclusions
This paper has at.lo~rlpf.cd 1.0 fi)cus or] our cl~rrcnl, 1c:vcl of l~ndcrst.anding of thc physical and chemical processes which influence the initiation and growth to detonation in granular, condensed phase explosives. T h e existing models a r e well-founded from a thermodynamic and mechanical point of view. However, the description of the chemistry is phenomenological, a t best, and certainly does not reflect a fundamental knowledge of the reaction mechanisms. This is readily apparent from our observations that, in general, the parameters describing the thermal decomposition of t h e explosive have n o bearing on those governing the thermal decomposition of hot spots a t high pressures. Clearly, we have been able to establish a set of parameters for both the hot. spot decomposition and the grain burning which suffice t o replicate the available experimental d a t a within the context of a multiphase mixture theory. Unfortunately this set of parameters may not b e unique and, in a more serious vain, the proposed reaction mechanisms may not even be appropriate. For example, it has been recognized for some time t h a t the grains fracture under shock loading anti, thus, it is possible t o envision chernical 1)onds being broken as the grains arcL cl(!avc,tf ros~ll(.ir~g i n fr.c,cs raclicals o r iol~s on grain sc~r.fa,ccs. Such pher~om- ena would have a profou~id infjuence on the model chosen t,o represent the surface reactions.
Chemical bonds may also be broken in zones of high shear within a grain. Indeed, it has been suggested t h a t it is these sisson reactions which are t h e cause of the rapid rollover t o detonation observed in the late stages of shock wave grow1.h. T h e current phenomenological models have difficl~lty in reproducing this result. These comments clearly point to the need for more detailed investigations of t h e fundamental chemistry of explosives a t high dynamic pressures. Efforts t o identify intermediate species which evolve during t h e loading conditions of interest could prove t o be extremely useful. It is hoped tha.t the recent advancements in optical diagnostics, including
:.pectroscopy, and quantum chemist.ry wili ~ h c d n f v l light on which species are present a t the
\-i;.rjous stages of wave growth and the rnitjor reaction paths.
Acknowledgement. - This work ~ e r f o r m e d a t Sandia National Laboratories supported by the I!. S. Department of Energy under contract number DE-AC04-76DP00789.
R e f e r e n c e s
[ I ] Bowden, F. P., and Yoffe, A. D., Initiation and Growth of Ezplosions i n Liquids and Solids (Cambridge University Press, Cambridge) 1952.
[2] Campbell, A. W., Davis, W. C., Ramsey, J. B., and Travis, J . R., Phys. Fluids 4 (1961) 551.
[3] Howe, P., Frey, R., Taylor, B. and Boyle, V., in Sixth Symposium (International) on Detonation (ONR) 1976.
[4] Wackerle, J., Rabie, R. L., Ginsberg, M. J., and Anderson, A. B., in Behavior of Dense Media under High Dynamic Pressure (CEA, Paris) 1978.
[5] Cochran, S. G., Lawrence Livermore National Laboratory Report UCID-18548 (1980).
161 Lee, E. L., and Tarver, C. M., Phys. Fluids 23 (1980) 2362.
[7] Johnson, J. N., Tang, P. K., and Forest, C. A., J. Appl. Phys. 57 (1985) 4323.
[8] Kipp, M. E., Nunziato, J . W., Setchell, R. E., and Walsh, E. K., in Seventh Symposium (International) on Detonation, J . M. Short, ed. (NSWC, White Oak) 1981.
191 Ilayes, D. H . , in Shock fi'cll~c1.s, fiplosiolls, c~nd I)elon,ntio?ts. .I. I?. nowen, el al, eds.
( A I A A , New York); l'rogress Astro. Aero. 87 (1983) 155 .
[lo] Hayes, D. B., Kipp, M. E., and Nunziato, J. W., in Shock Waves i n Condensed Matter- 1989, J . R. Asay, et al, e d ~ . (Elsevier, New York) 1984.
[ll] Nunziato, J . W., in Shock Waves i n Condensed Matter-1983, J . R. Asay, et al, eds.
(Elsevier, New York) 1984.
1121 Kipp, M. E., in Eighth Symposium (~nternational) on Detonation, t o be published (preprinted 1985).
1131 Baer, M . R., and Nunziato, J . W., Sandia National Laboratories Report SAND87-1411 (1987).
1141 Rogers, R . N.. Th,ernoehim. ,4cla 11 (1975) 131.
1151 Griffiths, N., and Groocock, J . M., J. (:hem. Soc. (London) 814 (1960) 4154.
[16] Price, D., and Bernecker, R.. R., inRehavior of Dense Media under High Dynamic Pressure ( C A E , Paris) 1978.
1171 Campbell, A. W., in Proc. JANNAF Propulsion System Hazards Meeting (Los Alamos, NM) 1980.
1181 Baer, M. R., and Nunziato, {J. W., Intl. J. Mulliphase Flows 12 (1986) 861.
JOURNAL DE PHYSIQUE
[I')] I3t>ggs, T. L., in Fundamentnls of Solid-P~ope1lrin.t Combustion, K . Kuo and M. Summer- field, eds., (AIAA, New York); Progress Astro. Aero. 90 (1984) 121.
i201 Sandusky, H. W., and Bernecker, R. R., in Eighth Symposium (International) on Deto- nation, t o be published (preprinted 1985).
1211 Von Holle, W. G., in Fast Reactions in Energetic Systems, C. Capellos and F. Walker, eds., (Reidel, Boston) 1980.
1221 Dick, J. J., Comb. Flame, 54 (1983) 121.
[23] Sheffield, S. A., Mitchell, D. E., and Hayes, D. B. in Sizth Symposium (International) o n Detonation, D. J. Edwards, ed. ( O N R , Arlington) 1976.
1241 Hayes, D. B., and Mitchell, D. E., in Behavior of Dense Media Under High Dynamic Pressure (CEA, Paris) 1978.
1251 Setchell, R. E., and Taylor, P. A., in Dynamics of Shock Waves, Explosions, and Deto- nations, J. R. Bowen, et al, eds. (AIAA, New York); Progress Astro. Aero. 94 (1985) 350.
1261 Renlund, A., and Trott, W., in Eighth Symposium (International) o n Detonation, t o be published (preprinted 1985).
1271 Setchell, R. E., in Eighth Symposium (International) o n Detonation, t o be published (preprinted 1985); also, in A P S Topieal Conference in Shock Wave Physics, 1987.
1281 Lee, E., Hornig, H. C., and Kury, J. W., Lawrence Livermore National Laboratory Report
UCRL-50422 (1968).
1291 C:owprrthwail.e, M . , and Zwisl(,r, W. 11., Slnnford Re.search I r r s l i l ~ c t c I'obl. %lo6 (1974).
A p p e n d i x : C o n t i n u u m T h e o r y o f M u l t i p h a s e M i x t u r e s A.1. E q u a t i o n s o f M o t i o n a n d T h e r m o d y n a m i c s
Chemically reacting mixtures, such as porous or granular explosives without binders, will be assumed t o consist of two phases, a ( a = s,g); t h e solid granular reactant (s) and the interstitial gas/gas products (g). On some appropriate length scale, each phase may be viewed a s occupying every spatial point x in the body. Physically, however, this is not the case; t h a t is, each phase occupies a volume distinct from all the others. Thus, we assign t o each spatial point x a phase density 7, and a volume fraction 4,. T h e phasct density represents the mass of the a-th phase per unit volume of the a-th phase, and the volume fraction represents the fraction of space occupied by the a-th phase a t the point x . Since the entire volume of t h e mixture is occupied,
d., = 4 .
+
4r = 1 :and t h e d,ensity of the mixture is the sum of the partial densities p,:
P = C P o 1 Ps = 4,,7" .
T h e motion of each phase is characterized by the velocity
v, = v,(x, 1) , v, =/ V', I (A31
T h e general equations of motion and energy for each of the phases can be expressed as:
pat, = ua o.. Vv, + para + e i - m i . v , - c,+(e, - v:/2) , ('46)
where the backward prime denotes the time derivative following the motion of phase a:
In (A4)-(A7), "V" denotes t h e spatial gradient, c,+ is the mass exchange due t o chemical re- actions, b, is the external body force (e.g., gravity), m: is t h e momentum exchange resulting from t h e forces, such as drag, acting a t t h e interfaces between phases, ea is t h e internal energy (per unit mass), 7, is the external heat supply, and e;t is t h e energy exchange due t o the local heat transfer between phases and the work done a t the internal phase boundaries. Also, u, is t h e symmetric stress tensor and i t will prove useful to express u, in terms of the phase pressure pa a n d the shear stress S , :
If we a d d together the equations of motion and energy for each.phase, then we should recover t h e well-known equations of motion and energy for a single material. Indeed, this requirement imposes restrictions on the interaction terms:
T h e Sccarid 1,aw of t.hcrr1iodyrrarr1ic.5 svrvcs 1.0 establish rtistrict,ions on I'orrrrs of t,he consti- tutive equations appropriate for reactive mixtures. In terrrls of the Hel~nholtz free energy
where Ta is the absolute temperataure and s, is the entropy, then t h e Second Law can be expressed as the thermodynamic inequality
A.2 C o n s t i t i l t i v e M o d e l s and C l o s u r e
With t h e equations of motion and energy given, it remains t o consider the specific con- stit,ut.ive models which are appropriate for gra~lular energetic materials without hinders. The ir~dividrlal p11asc.s arc: c:lrarly sciparal.cd pl~ysically ar~tl 1.11us, it, is pla~lsibl(: 1,o ~.cclrlirc, thab the:
therrriodynarriic variables for a given phase depend only on the state variables y,, v,, $,,T, corresponding t o t h a t phase. In particular, for the solid reactant, a nonlinear thermoelastic description of the Helmholtz free energy $J, is used which has proven t o describe a broad class of explosives 1231. Shear stresses are neglected in this dcscription and the constants are obtained from Hugoniot and thermophysical data. For the gaseous reaction products, the Jones-Wilkins- Lee (JWL) equation of s t a t e is employed which can describe the very dense, thermodynamic states encountered a t detonation as well as ideal gas behavior at low pressures and tempera- tures 1281. The constants for this model are best, fit to d a t a and hydrodynamic calculations of
