9786
J . Phys. Chem. 1994, 98, 9786-9791
Energy Transfer Rates in Primary, Secondary, and Insensitive Explosives Laurence E. Fried* and Anthony J. Ruggiero L-282, Energetic Materials Center, Chemistry and Materials Science Department, Lawrence Livermore National Laboratory, Livermore, California 94550 Received: November 15, 1993; In Final Form: June 15, 1994@
In this paper we focus on the relation between the propensity of a molecular crystal to undergo chemical reactions after impact or shock and energy transfer rates. When a crystal receives a shock, low-frequency lattice vibrations (called phonons) are excited. Typical phonon frequencies are 0-200 cm-’. This energy must then be converted to bond stretch frequencies (1000-2000 cm-’) before bond breaking can occur. W e derive a simple formula for the total energy transfer rate into a given vibron band in terms of the density of vibrational states and the vibron-phonon coupling. We are able to estimate the phonon upconversion rate in widely varying energetic materials such as TATB, HMX, and Pb styphnate by examining existing inelastic neutron scattering data. We find that the estimated energy transfer rates in pure unreacted material are several times greater for the sensitive explosives studied than the insensitive explosives.
The initiation of a crystal upon shock is a complicated process undoubtedly depending on a host of material properties. Upon shock, low-frequency lattice vibrations called acoustic phonons are primarily excited. Before a detonation wave can begin, bonds must break. Therefore, the initial energy in the acoustic phonons must somehow be deposited into bond-stretching modes. Acoustic phonons have frequencies less than 100 cm-’, whereas bond stretches have frequencies greater than 1000 cm-’. It is clear, then, that acoustic phonon energy must be upconverted to higher vibrations before detonation can occur. Highfrequency vibrations in a molecular crystal are called vibrons. Dlott and Fayer9.10 have studied multiphonon upconversion associated with shock-induced chemistry. They derived a master equation for phonon and vibron temperatures. The master equation depends on the phonon-vibron energy transfer rate, which was estimated from experimental results on anthracene. Kim and Dlott11,12have studied multiphonon up-pumping in naphthalene through molecular dynamics and the master equation approach. There has been much work on deriving theoretical expressions for phonon lifetimes in anharmonic solid^.'^-'^ Califano et al. compared the theoretical expressions to the results of molecular dynamics calculations and experiment~.’~-’’Holian19 has conducted molecular dynamics simulations of vibrational energy transfer in diatomic fluids, with the aim of understanding shock-induced chemistry. In the present paper, we address the question of how the phonon-vibron energy transfer rate differs in a wide range of energetic materials. Our ultimate goal is to understand how phonon dynamics controls chemical reactivity. As mentioned above, a chemical reaction is the end result of a long chain of events. Phonon energy must be upconverted to higher frequencies (vibrons), and then vibron energy must be localized in a particular molecule. Since defects in solids (often micron-sized voids) produce localized regions of high temperature, we expect that a treatment of defects (defects which lead to enhanced ignition are called “hot spots” in the explosives community) is necessary to understand the localization step. Finally, localized vibrational energy in a molecule must be channeled into a reaction coordinate. Reaction occurs when the excitation of the reaction coordinate exceeds the dissociation energy. At the present time, we do not have the information necessary
’-’
@Abstract published in Advance ACS Abstracts, September 1, 1994.
to evaluate phonon-induced reaction rates for a wide range of energetic materials. Progress can be made, however, by studying isolated parts of the chain of events described above, with the intention of putting the parts together as more information becomes available. In the present work, we will study how efficiently various energetic materials upconvert phonons into vibrons. This requires a microscopic formulation of the phonon-vibron energy transfer rate. Since optical experiments probe states with crystal momentum k =Z 0, in the past attention has been focused on the lifetimes of k x 0 states. Chemical reactions, however, do not have this selection rule. This motivates studying the upconversion rate into states with arbitrary k, an entire band rather than the k =Z 0 edge of the band. Furthermore, it is reasonable to suppose that certain vibron modes will be more efficient in driving a chemical reaction than other modes. Hence, we will fix the energy of the final (vibron) state w3 and sum over all phonon states (w1, w2) such that w1 w2 = 0 3 . This is in contrast to the phonon lifetime problem, where the initial phonon energy w1 is fixed and w2 and w3 are varied. The frequency dependence of the upconversion rate is found to be very important; in the range 100-600 cm-I, the rate can vary by an order of magnitude. This means that a quantitative theory of phonon upconversion cannot be based on a single vibron-phonon rate, but rather should be based on a temperature and frequency dependent rate. Since shock sensitivity data is not available for the wide range of explosives under study here, we compare our energy transfer rates to impact sensitivity data. We find that the predicted upconversion rate is much higher for sensitive explosives (with sensitivity measured by the standard drop hammer test) than for insensitive explosives. This suggests that the upconversion step of the chain of events leading to reaction could possibly be responsible for many of the differences observed between different materials. Dlott and Fayer have suggested that anharmonic defects in a crystal could play an important role in initiation; our results suggest that anharmonicity not associated with a defect can also play a role. This type of anharmonicity provides a pathway for the excitation of high-frequency modes in regions of defect-free material. Our results also suggest that many aspects of phonon-induced reaction rates can be attributed to properties of the pure, defectfree unreacted material.
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0022-365419412098-9786$04.50/0 0 1994 American Chemical Society
Energy Transfer Rates in Explosives
J. Phys. Chem., Val. 98, No. 39, 1994 9787
1. Theory We consider phonon upconversion rates in the limit of small anharmonicity (e.g. cubic terms in the Hamiltonian are assumed to dominate). In this case, the Hamiltonian is given by
H = H, + H, Here, a,b labels the bands of the initial states, while c is the band of the final state. V is the volume of the crystal. As discussed above, we are interested in the rate for upconversion into a given band averaged over all k,. Integrating over k,, we have
+
where AI a1 art, and a1 is defined to be the annihilation operator for phonon mode 1. W I is the harmonic frequency of phonon 1. A(k) = 1 if k = 0 and is 0 otherwise. N is the number of atoms in the crystal. Phonon lifetimes have been extensively studied using thermal Green functions13 and other theoretical techniques.I6 Part of the decay rate of a phonon comes from scattering to higher frequencies; this term is called the phonon upconversion rate. In the present paper we consider a related but distinct quantity. In phonon lifetime studies the relevant rate is the collision of a pre-existing k = 0 phonon of a fixed frequency with a thermal phonon of arbitrary frequency. Here, we consider the collision of two thermal phonons with a fixed energy sum but arbitrary k. This allows us to study the rate of energy transfer into a particular vibron band. Certain vibron modes are likely to lead directly to molecular dissociation, whereas others do not. The rate of energy transfer from thermal phonons into a mode that causes dissociation should be most relevant for initiation. Phonon lifetimes in crystals are usually derived by calculating the imaginary part of the phonon self-energy using thermal Green functions. In the present case it suffices to use Fermi's Golden Rule. The state to state energy upconversion rate y for the fusion of two phonons of frequency w~and wminto a phonon of frequency w n is
where n, is the occupation number of phonon j . Equation 2 is easily evaluated to be 18n y(n,l,m) = -(n Nh2
l)nln,IB,,12d(w,
- w, - w,)A(k,
-
k, - k,) (3) Here, kj is the wave vector of phonon j ; wj is the frequency of phonon j . For upconversion processes, it is reasonable to assume a thermal distribution of phonons 1 and m,with state n unpopulated,20 since low-frequency phonons are primarily excited by impact and low-frequency phonons equilibrate faster (i.e. have shorter lifetimes) than high-frequency phonons. This gives a thermal rate of 18n- y(n,l,m,T) = -nn IBImn12d(wn - w, - w,)A(k, - k, Nh2
'
k,)
These equations can be substantially simplified through two approximations. Since vibrons typically have little dispersion,16 we can make an Einstein approximation for the vibron band: w&fkb) = w . Also, the mode dependence of B is often weak.I6 Thus, we let B(a,k&kb,c,k,) = B. Comparison with two phonon absorption line shapes shows that this approximation is usually very good.I6 After integrating over k,, we arrive at a simple expression for the upconversion into a band of a given energy per volume of crystal:
Here, ~ ( wis) the density of states, defined by
(9) Equation 8 should be evaluated for discrete w , corresponding to a band frequency. It is possible, however, to treat w, as a continuous variable. This is motivated by the high density of vibron bands found in the large molecules considered here. The results derived in this fashion are physically meaningful as long as there exists a real vibron frequency in the neighborhood of WC.
Optical probes only yield information on the density of states near k = 0. The total density of states e ( w ) can be best estimated from incoherent inelastic neutron scattering. The procedure involved is outlined in ref 2 1. The density of states derived from neutron scattering data is known only to within an arbitrary normalization factor. In order to compare energy transfer rates from different molecules, it is necessary to determine this normalization factor. The simplest altemative is to take the integral of @ ( w )over all frequencies to be equal to 3NIV, where N is the number of atoms in the crystal and V is the volume. Existing experimental data, however, are not available at high enough frequencies to make such a normalization feasible. Instead, we employ a simple normalization rule proposed by Dlott:22 we normalize e ( w ) so that
(4)
where T is the temperature and
Next, we average over all initial states 1 and m. This gives
Here, QP is the maximum frequency of the phonon bands, Z is the number of molecules per unit cell, and Y is the number of molecular vibrations that have become amalgamated into the phonon band. V, is the volume of a unit cell. (The V , factor
9788 J. Phys. Chem., Vol. 98, No. 39, 1994
3 Pb Styphnate
3
Y
a
O
400
200
600
w (cm-’1
3 Picric acid
Fried and Ruggiero TABLE 2: Thermodynamic Data (Ref 24) and Griineisen Parameters ( Y ) Calculated with Ea 11 compound
e-
(kg/m3) RDX 1806 PBX-9404(94%HMX) 1840 LX-17 (95% TATB) 1899
cb-
(W
a(K-9
2650 6.3 x 2260 5.8 x 2240 6.0 x
c,-
(JkW
Y
1126 1130
1.13
0.78
1130
0.79
no gap in the density of states between vibron modes and collective phonon modes. This is consistent with the large number of low-frequency vibrations found in the MOPACPM3 calculations of Table 1. The lack of a gap in the vibrational density of states implies that these molecules should upconvert phonon energy much faster than stiffer molecules (e.g., diatomics). The compounds in Figure 1 are listed in order of decreasing impact sensitivity. The most sensitive compounds (Pb styphnate, y-HMX) have a higher density of states than the least sensitive compound (TATB) for w > 100 cm-’, with the biggest difference noticeable in the 400-600 cm-I region. The density of states of TATB also is “flatter” than that of Pb styphnate or y-HMX. We expect energy transfer rates to be more closely related to sensitivity than the density of states alone. We have evaluated the energy transfer rate in eq 8 by assuming that the anharmonic coupling per molecular volume is the same for all the compounds considered here; this is equivalent to taking B = V,Bd Z . We can check this approximation in several cases by calculating Griineisen parameters. The Griineisen parameter
P
1
TATB
0
400
200 w
600
(cm-’)
Figure 1. Densities of states determined by inelastic neutron scattering. TABLE 1: Number of Amalgamated Vibrations As Calculated with MOPACPM3 and Drop Hammer Test Results for a 12 Tool molecule p - m x
y-mx
Pb styphnate
picric acid RDX
styphnic acid TATB
Y 12 15 12 8 7 8
7
hso(m) 0.33 (ref 24) 0.14 (ref 25) 0.15 (ref 26) 0.73 (ref 24) 0.28 (ref 24) 0.73 (ref 26) 3.20 (ref 24)
was added to the original normalization rule of Dlott to maintain consistency with eq 9.) We have taken QP = 200 cm-’, while Y was estimated as the number of vibrational modes of the isolated molecule with frequencies less than 52,. The vibrational modes were calculated with the semiempirical electronic structure program MOPAC using the PM3 H a m i l t ~ n i a nresults ; ~ ~ are given in Table 1 along with impact sensitivity m e a s u r e m e n t ~ . ~ ~ - ~ ~ 2. Results We have used existing inelastic incoherent neutron scattering data21,27to determine the density of states e(@),as described above. Figure 1 shows g(w) found by this procedure. The densities of states are all remarkably smooth. This is typical of “floppy” molecules such as durene. In particular, there is
is a measure of the overall anharmonicity of the where is the bulk modulus, a is the coefficient of thermal expansion, is the density, and C, is the constant volume heat capacity per unit mass. In evaluating the Griineisen parameters, we have made use of the approximate relations C, C, and B x gcb2, where C, is the constant pressure heat capacity and cb is the bulk sound speed. The necessary thermodynamic constants are known for RDX and formulated versions of TATB and HMX. As shown in Table 2, the Griineisen parameters are similar, indicating that the anharmonicity in these crystals is roughly the same. It is also important to note that the anharmonicity can have a frequency dependence; usually this dependence is weak.I6 Figure 2 shows the upconversion rate as a function of frequency and temperature for the explosives studied. All the results are shown at the same scale, except for y-HMX. Experimental data on the density of states are only reliable up to about 600 cm-’, so we chose that to be the upper limit of the calculations. Low-frequency results (0-200 cm-’) are provided for comparativepurposes; the approximations involved in deriving eq 8 should work best for frequencies above 200 cm-1. The energy transfer rate depends strongly on temperature. Obviously, at higher temperatures the upconversion rate increases due to a higher population of phonons. When kBT >> w,the energy transfer rate depends quadratically on temperature: Bb
w),kBT >> 0,(12) This quadratic scaling is evident in the plots, even when ~ B isT not greater than w. This is because it is possible to have the integral over w in eq 8 dominated by w
