A theoretical equation of state for detonation products is described that places particular emphasis on the characterization of small carbon clusters (20{angstrom}--50{angstrom} in diameter) in the products. Diamond clusters are modeled with the dangling bonds on the surface atoms (up to 30% of the cluster) capped by various radicals composed of C, H, N, and O from the background molecular fluid mixture. Free energy methods for the surface groups are used to determine the chemical equilibrium composition of the cluster surface as well as the surrounding molecular fluid mixture. The surface composition shows dramatic changes in composition over some regions and varies slowly in others. A perturbation theory approach is used for the mixture of molecular fluids that also includes features based on Monte Carlo simulations

Shaw, M.S.

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A theoretical equation of state for detonation products

Shaw, M. S.

C * /  LA-UR-992 3 3 2 4  Title: Author(@: Submitted to: A THEORETICAL EQUATION OF STATE FOR DETONATION PRODUCTS WITH CHEMICAL EQUILIBRIUM COMPOSITION OF THE SURFACE OF SMALL CARBON CLUSTERS M. Sam Shaw, T-14, MS B214 Los Alamos National Laboratory Los Alamos, New Mexico 87545 APS Topical Conference on Shock Compression Condensed Matter Snowbird, Utah June 27 - July 2, 1999 Los Alamos N A T I O N A L  L A B O R A T O  R Los Alamos National Laboratory, an affirmative actlordequal opportunity employer, is operated by the University of California for the US. Department of Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher recognizes that the US. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or to allow others to do so. for US. Government purposes. The Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Form No. 836 R5 ST2629 10/91 A THEORETICAL EQUATION OF STATE FOR DETONATION PRODUCTS WITH CHEMICAL EQUILIBRIUM COMPOSITION OF THE SURFACE OF SMALL CARBON CLUSTERS * M. Sam Shaw Group T-14 MS B214, Los Alamos National Luboratory, Los Alamos, New Mexico 87545 USA Abstract. A theoretical equation of state for detonation products is described and compared with various data. A perturbation theor approach is used for the mixture of molecular fluids that is based on Monte Carlo of the clusters is modeled as primarily additive contributions such as vibrational modes, bond strengths, and effective volumes. Up to 30% of the atoms in a diamond cluster are on the surface with dangling bonds cap ed by various groups composed of C, H, N, and 0 €rom the background fluid. A counting term similar to &ea1 entrop of mixing is also found for the surface composition. Competition between the U, TS, and PV terms in the Gi  bs free energy leads to dramatic shifts in the surface composition in some regions. This in turn leads to shifts in the background fluid mixture composition and anomalous behavior in the total EOS. This behavior in PBX 9502 is in good agreement with recent data on release isentropes from overdriven states( 1). simulations. The solid car il on IS characterized as small clusters rather than as a bulk material. The free energy INTRODUCTION The detonation products of a high explosive form a very nonideal and complicated mixture under ex- treme conditions (P NN 30 GPa and T M 3000 K). An accurate theoretical treatment must not only handle well the extreme conditions for a single molecular fluid, but also the mixture of species and the ther- modynamic chemical equilibrium composition of the products. In addition, the solid carbon is in the form of small clusters with a large fraction of atoms on the surrface. In the diamond phase, the spherical clusters range from 20 - 50 8, in diameter with 20 - 30 96 of the atoms on the surface with energetically unfavor- able dangling bonds. In this work, we provide for a variety of dangling bond cap groups in chemical equilibrium. The back- ground fluid treatment is given by a fluid perturbation theory tied to benchmark simuation methods. CARBON CLUSTERS In this section we focus on the treatment of dia- mond clusters. A more detailed description can be *This research is supported by the Department of Energy, un- 1 der contract W-7405-ENG-36. found in a separate paper(2). (Graphite has no dan- gling bonds except at the edges and is treated as a bulk graphite EOS with a small shift in the heat of formation to include restructuring at the edges.) A spherical diamond cluster has dangling bonds ex- posed over the entire surface. It is energetically unfa- vorable to restructure the diamond surface(3). Figure 1 shows a slice through the z=0.5 (lattice constant) plane for a 23 8, cluster. A mirror plane is indicated as a dashed line. Surface atoms with single and dou- ble dangling bonds (denoted 1 and 2) are shown in the upper left. Examples of possible surface groups are shown in the corresponding mirror sites in the lower right. Nonsurface atoms, denoted 'core car- bon atoms', are indicated by a circle and the origin is given by a plus. Also note that the tetrahedral bonds are all out of the plane and none of the atoms in a plane are bonded to each other. For a given radius cluster, the number and type of surface groups in the truncated lattice and the number of core atoms without dangling bonds are counted. Figure 2 shows the ratio of surface to core atoms as a function of the number of core atoms for discrete possible clusters. The line shows an analytic fit that is 2 1  / 2 0 0 0 I:/ 0 0 0 0 / ' / O  0 0 0 O / / O  0 0 0 O / J O  0 0 1 0 O / O  0 0 CNH 0 O / / O  0 0 0 'H Q / / O  0 0 0 2 0 0 0 0 ,/'N 2 0 0 O . / / O  0 N c"co l / O  . o  0 0 , I /  c 0 0 0 c, ' H  c c  0 \\ H CH* FIGURE 1. The ~ 0 . 5  plane of a 23 8, cluster. A mirror plane is indicated as a dashed line. Surface atoms with single (1) and double (2) dangling bonds in the upper left. Examples of surface groups in the corresponding mirror sites in the lower right. dominated by the geometric surface to volume ratio of a sphere. The relative fraction of single dangling bond sites is nearly constant at about 0.65 over a wide range of cluster size. In order to determine the equilibrium composition of the surface, it is necessary to determine the Gibbs free energy of the cluster with arbitrary surface com- position. Figure 3 shows a diagram of a small part of the surface. Each surface group has an additive contribution to the free energy. An Einstein model is used for the vibrational frequencies in surface group. These modes are assumed to couple weakly to the core and are estimated from similar small molecules where possible. The heat of formation of each sur- face group is determined by additive bond strengths. The effective volume is estimated from the geome- try of similar groups in small molecules. An effec- tive standoff distance from the geometric surface is chosen to take into account that the occupied volume (Le. volume excluded from the molecular fluid) is not identical to the geometric volume. Notice that a I 0 6000 10000 0.0 I NCO, FIGURE 2. Ratio of surface sites to core atoms versus the number of core atoms in a cluster substitutional N atom would occupy a small volume that will become significant in the PV term at high pressure. The surface free energy has an entropy contribu- tions due to the counting of possible ways to occupy a set of surface sites with a given composition. This contribution to the Helmholtz free energy for single bond sites is given by where zi(1) is the concentration (mol) of surface groups of type i and dl) is the total concentration of single dangling bond sites (and similarly for the double bond sites). Finally, the core contributions are taken from a bulk diamond model incorporating a Debye model of vibrations and a measured cold curve. A related model by van Thiel and Ree(4) implemented simi- lar characterization of the surface groups with a fixed choice of surface composition. FLUID MIXTURE Benchmark simulation methods are used to test the various approximations used to evaluate the fluid mixture EOS. Accurate fluid perturbation theories PBX 9502 data (0) equlllbrlum (line) no N (dot) no H (dash) 100.0 80.0 cluster I surface I r l  1 FIGURE 3. Diagram of the effective surface and occupied vol- ume (box) of additive surface groups and other approximations are used for spherical po- tentials, nonspherical potentials, nonideal mixing, and chemical equilibrium composition. Details are given elsewhere(2) because of space limitation. The parameters in this method were least squares fit over a limited set of data and with terms added to limit the variation of the constants where an es- timate could be made of the value and uncertainty. The slope change in the overdriven Hugoniot of PBX 9502 had a strong influence(2) on the final set of pa- rameters. Further refinement is planned with a much larger data set, in particular using more of the new types of measurements such as sound speeds(5) and temperatures(6). RESULTS Vorthman et al.(l) have measured the release isen- trope in PBX 9502 from overdriven states. The anomalous behavior of the data at high pressure is shown in Fig. 4. The present theory is in very good agreement with the data and provides a possi- ble mechanism for the anomalous behavior. At high pressure, the single dangling bond surface C atoms are replaced by a substitutional N atom with no dan- gling bonds. The PV term gives preference for the small volume occupied by this surface group. At low pressure, the lower energy of a H cap on the dangling 0.0 I 2.0 2.5 3.0 P W ~ ’ )  5 FIGURE 4. PBX 9502 isentrope from a high pressure over- driven state. Data (01, Theory: equilibrium (line), substitutional N (dash), H cap (dot). bond dominates. The transition in surface chemistry is spread over 20 GPa in pressure and 0.5 g/cm3 in density. The theory reproduces the rapid slope change around 50 GPa. In the transition region, the slope of the isentrope is very nearly constant. That implies the sound speed is also nearly constant over that region. Figure 5 shows the rapid variation in sur- face composition that produces the unusual EOS. There is also a very large effect due to the graphite diamond transition. Figure 6 shows good agree- ment between theory and experiment(7) for a series of detonation velocity versus density for TNT, 25/75 TNTRDX, 50/50 TNT/RDX, and RDX. The deto- nation velocities are dominated by the graphite di- amond transition. Although the transition region is rather small (1 - 2 GPA), Fig. 7 shows that the mini- mum shock velocity is sometimes determined by the location of the equilibrium transition rather than the minimum of either phase. CONCLUSIONS The chemical equilibrium treatment of surface chemistry on diamond clusters was started based on the premise that there was too large a fraction of atoms on the surface to treat the carbon as a pure bulk material. The calculations demonstrate that there can PBX 9502 COMPOSITION OF SINGLE DANGLING BOND SURFACE Y 0 oc 3 0.4 8 SITES P (GPa) FIGUIkE 5. Composition of single dangling bond surface sites. Moles of surface group per formula weight versus pressure on the isentrope shown in Fig. 6. FIGURE 6. Detonation velocity versus density for TNT, 25/75 TNTIRDX, 50/50 TNTIRDX and RDX. Theory: equilibrium (line), graphite only (chain dash), and diamond clusters only (dash). 7.5 ::: i1.2 1.4 1.6 1.8 2.0 Up (km/s) FIGURE 7. Shock velocity versus particle velocity at two values of initial density of RDX in the transition region, Equilibrium (line), graphite only (chain dash), and diamond only (dash). CJ state for equilibrium (O), graphite only (+), and diamond only (*) periments. These preliminary results suggest that fit- ting an EOS theory to detonation velocity data is not as straightforward as it seems. Significant portions of the data could be strongly influenced by transition regions in the chemical composition. REFERENCES 1. J. E. Vorthman, W. W. Anderson, J. N. Fritz, R. S. Hixson, and M. S. Shaw - this conference. 2. M. S. Shaw, "A Theoretica Equation of State for Detonation Products", Eleventh Symposium (International) on Detona- tion, to be published, 3. N. W. Winter and F. H. Ree, "Stability of the Graphite and Dia- mond Phases of Finite Carbon Clusters", Eleventh Symposium (International) on Detonation, to be published. 4. M. van Thiel and E H. Ree, "Nonequilibrium Effects of Slow Diffusion Controlled Reactions on the Properties of Ex- plosives", Ninth Symposium (International) on Detonation, 5 .  Schmidt, S. C., Moore, D. S., and Shaw, M. S., J.  Chem. Phys. 6 .  Fritz, J. N.,Hixson, R. S., Shaw, M. S., Morris, C. E., and Mc- 7. Dremin, A. N., Pershin, S. V., Pyaternev, S. V., and Tsaplin, D. pp.743-750 (1989). 107,325 (1997). Queen, R. G., J. Appl. Phys. 80,6129 (1996). N., Fizika Goreniya i Vzryva 25, 141 (1989). be regions with a large shift in the chemical com- position on the surface and that the consequences of these shifts arc within the range of sensitivity of ex- A THEORETICAL EQUATION OF STATE FOR DETONATION PRODUCTS WITH CHEMICAL EQUILIBRIUM COMPOSITION OF THE SURFACE OF SMALL CARBON CLUSTERS M. Sam Shaw, T-14 Los Alamos National Laboratory 1999 TOPICAL CONFERENCE O N  SHOCK COMPRESSION OF CONDENSED MATTER June 28 - July 2 1999 Snowbird, Utah OVERVIEW DETONATION PRODUCTS Unreacted: c,HbN,od + binder Products: dense fluid mixture ( 7 0 2 ,  N2,1120, CO, + minor species carbon clusters z 1000 atoms z 20 gr ap hit e-li ke diamond-like APPROACH 0 Characterize Cluster Surface 0 Chemical Equilibrium of Surface e Simulation Methods 0 Perturbation Theory e Constrained Fit mirror plane CROSS SECTION OF A SMALL DIAMOND CLUSTER CORE ATOMS: 0 = 0 DANGLING BONDS SURFACE ATOMS: 1 = 1 DANGLING BOND 2 = 2 DANGLING BONDS 2 1 / / / 2 0 0 0 1 .  / 0 /’ N 2 0 0 0 0 0 0 0 0 0 0 / /  0 0 2 0 0 0 / /  0 0 N / 0 / /  0 / / 0 0 O f 0  0 0 0 0 / /  0 0 0 CRH / 1 0 0 / /  0 0 0 ‘H c / 0 / /  0 / 0 0 0 0 i //o 0 0 0 C +co / r  ’/C 0 0 0 / /  H C C EXAMPLES OF SURFACE GROUPS note: N without dangling bonds can substitute for a surface C with one dangling bond 0 0 0 iu 0 core N surface /N 0 b 0 is, 0 cn 0 0 0 0 0 0 iu Fraction of Surface Atoms 0 b 0 b 0 00 -- e $r 3= rc -- 1- cluster I surtace ..... ..... .. I...WU..I.I" I'. .A 7 ....... " . r  ..... U... I I H H $=C ~ NIY I surface De by e I t '  A I-i- I - - -' \ I 100.0 80.0 n cb n (3 60.0 a. U 40.0 OVERDRIVEN HUGONIOT PBX-9501 (0) PBX-9502 (+) I . I . I . 1 I I 2.5 3.0 3.5 4.0 20.0 2.0 a 100.0 80.0 60.0 40.0 20.0 PBX 9502 data (0) equilibrium (line) no N (dot) no H (dash) 3.5 1 .o 0.8 0.6 0.4 0.2 0.0 0 PBX 9502 COMPOSITION OF SINGLE DANGLING BOND SURFACE SITES I I 20 40 P (GPa) 60 80 9.0 1 I I r / / / / / + / , /'+ ' c ?F a 9.0 8.5 n m \ E Y 8.0 e7.5 50/50 RDXrrNT Equilibrium (line) Diamond (dash) Graphite (chain) -1- I I I I I I I '. '. 7.0 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 Up (km/s! 