The Journal of
Physical Chemistry
0 Copyright 1994 by the American Chemical Society
VOLUME 98, NUMBER 43, OCTOBER 27,1994
LETTERS Defect-Enhanced Structural Relaxation Mechanism for the Evolution of Hot Spots in Rapidly Compressed Crystals D. H. Tsai* 10400 Lloyd Road, Potomac, Maryland 20854
R. W. Armstrong Department of Mechanical Engineering, University of Maryland, College Park, Maryland 20742 Received: March 9, 1994; In Final Form: June 21, 1994@
The evolution of hot spots at defects in a 3D fcc lattice under rapid 1D compression along different crystallographic directions has been studied by mean of molecular dynamics. Hot spots were generated in monatomic and molecular crystals when the strain energy released through defect-mediated structural relaxation was locally converted to heat. Thus, hot spot temperatures are shown to be controlled by the potential energy source of the strained lattice and by the allowed extent of structural relaxation.
Introduction The initiation and sensitivity of energetic materials are important topics of current research. In a recent symposium volume’ on energetic materials, for instance, fully one-third of the papers dealt with one or more aspects of these problems. In a report given there on a molecular dynamics (MD) study of hot spots in 2D crystals under rapid (and shock) compression,2 attention was directed to the fundamental role of structural relaxation around defects in the heating of hot spots. In ref 3, it was demonstrated that the effective stress-strain relation in a 2D bcc MD model, containing a 10-atom vacancy cluster, showed a “yield stress” under rapid compression when the vacancy cluster collapsed into a residual dislocation dipole network, accompanied by only minor movement of the dislocations. The previous work has now been extended to a 3D model of an fcc crystal, monatomic or molecular, containing defects such as vacancies, grain boundaries, impurities, etc., and similarly subjected to 1D compression in different directions with respect to the crystal axes. The results show that, under @Abstractpublished in Advance ACS Abstracts, October 15, 1994.
load, local relaxation of the structure is triggered by the onset of instability, and the relaxation is accompanied by conversion of a part of the local strain energy to kinetic energy. Thus, the creation of hot spots in rapidly compressed or shocked crystals may be considered, quite generally for 2D and 3D model crystals, to result from the conversion of the potential energy source associated with the strain, through the mechanism of structural relaxation, to kinetic energy and local heating. In this Letter, further emphasis is given to the relative importance of the energy source(s) and to the fundamental role of structural relaxation in providing the mechanism for energy conversion. The connection to a special “hollow core” characteristic proposed for dislocations in energetic materials and to certain aspects of macroscale experiments is also discussed.
Model The MD model is a 3D fcc lattice in Cartesian coordinates, with the Z axis in the [OOl] direction and the X axis at 0”, 14.04”, and 45” from the [loo] direction, the last being the [110] direction. At 0’ and 14.04’, the system consists of 32 x 16 x 16 (X,Y,Z) lattice planes (1024 unit cells, 4096 atoms); at 45’,
0022-365419412098-10997$04.50/0 0 1994 American Chemical Society
Letters
10998 .IPhys. . Chem., Vol. 98, No. 43, 1994 the system consists of 40 x 20 x 20 lattice planes (1000 unit cells, 4000 atoms). For the monatomic lattice, the two-body interaction potential is a Morse potential V with a well depth A = 1. V has a cutoff at an interatomic distance R = 1.6 between atoms i and j . For 1.6 5 R 5 1.8, Vis joined to the zero energy axis by a third degree polynomial in R, with continuous energy and first derivatives at R = 1.6 and 1.8. For the molecular lattice, A = 2 when both i and j belong to the same molecule and A = 0.5 when they belong to different molecules. Each molecule is made up of eight atoms arranged in the shape of a box with a square base. The molecules are packed into an fcc structure which proved to be stable. Low-level shock compression, of around 8- lo%, is of special interest here because this is the range in which structural relaxation may be expected to occur. But here, the compression is applied by shrinking the distance scales (X,,Y,,Z,) uniformly over the whole volume, at a rate corresponding to the transit time of a longitudinal sound wave through the distance of a typical defect cluster. This was done in the interest of reducing the amount of computation. Investigation4 showed that the hot spots obtained this way were comparable to those obtained by a shock wave propagating through the system, because the structural relaxation process was slow compared with the compression process. Other details of the model, including the makeup of vacancies and other defects, the algorithm for numerical integration, etc., are described in ref 2. A hot spot is defined as a region comprised of several atoms, with a higher kinetic energy Ek than the average of the system and a lifetime of several atomic oscillations so that the hot spot may be said to be in local equilibrium. Figure l a shows a superposition of two adjacent middle (001) planes of the molecular model at time t = 0, with the X axis in the [ 1101 direction and the atoms shown as open squares and crosses. These planes contain one vacancy (one missing molecule) at the center, the only defect in an otherwise perfect block of crystal with periodic boundaries. Figure l b shows the distribution of the potential energy E, per atom in the two planes. E,, (in the figure caption) is the average of the potential energy of the entire model. Epa (dashed outlines) is the average Ep, and Epmthe maximum, in these two planes. The raised potential energy surface at the center is the result of strain from the vacancy.
Results Figure 2 shows the changes in E, and Ek vs time when the lattice of Figure 1 is uniaxially compressed 8%, from X, = 0.4954 to 0.4558, at a uniform rate in the first 0.8 unit of time. Elastic Compression. In elastic compression of a solid, as distinct from compression of a gas, most of the work goes into the lattice as increased potential energy and only a small part of the work as increased kinetic energy. For example, in Figure 2, between t = 0 and 0.8, and before any structural change has occurred, AE,, is 0.627 unit of energy and only 0.012. If the potential energy distribution is not uniform, because of the presence of defects and/or differences in the inter- and intramolecular interactions, the more compressible region would experience greater temperature rise. But as long as there is no structural change, the temperature variation would remain small. Under these conditions, when Ek (Figure 2b) is averaged over several atoms (as for Ep in Figure lb), the maximum E h is no more than 2-2.5 times higher than Ek,, the same as when the system is in equilibrium and not compressed, with a MaxwellBoltzmann distribution of energies. The value of 2-2.5 may thus be used as a figure of merit for distinguishing a hot spot from the natural equilibrium fluctuations in the kinetic energy in the model crystal.
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Figure 1. (a) The initial configuration of the molecular model employed in this investigation. Two ( X , y ) lattice planes, no. 9 (0) and no. 10 (+), in a block of 20 planes in the [OOl] (2) direction, are shown, with the X axis in the [110] direction. The labels denote the lattice plane numbers. Each box-shaped molecule was made up of eight atoms, four in plane 9 and four in plane 10. The molecules in alternate pairs of planes were arranged so as to produce an fcc structure which proved to be stable. At equilibrium, the interplanar distances were X , = Y, = 0.4954 and Z, = 0.7023. At these values of (X,,Y,,Z) the normal stress components in the lattice were near zero at zero temperature. (b) Potential energy distribution in planes 9 and 10. The energy at each lattice site was the average of the energy of the atom(s) at that site and those at the 12 nearest-neighboringsites. The nubbly surface resulted from the different inter- and intramolecular interactions. For the whole system at equilibrium Eps= -6.426 and Eks = 0.057; for planes 9 and 10, EPa= -6.411, Epm= -5.694, Eka = 0.055, and E h = 0.094. The unit of energy was unity. Ep/Atom -4.0 j
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Figure 2. Epand & vs time t as the molecular lattice of Figure 1 was uniaxially compressed by reducing X , from 0.4954 to 0.4558 at a uniform rate in 0 . 8 ~E,,,, and E h refer to the maximum values of E ,
and E k that occurred at time t, not at a fixed location, but somewhere within the model. Between t = 14 and! = 20 the average energies of the system in quasi-equilibrium were E,, = -5.950, Epm= -4.875 and E k s = 0.219, E h = 0.444.
Energy Source from Structural Defects. The presence of a defect results in some local strain energy that is usually large compared with E b . In Figure lb, for example, AE, = 0.732 at the defect site, and Eks = 0.057 at t = 0. The strain energy is also increased substantially by mechanical compression, even for the low-level compression as noted in Figure 2. In addition, the defect tends to reduce the mechanical strength of the structure. Thus, if the compression should exceed the local stability of the structure, structural relaxation would likely start
J. Phys. Chem., Vol. 98, No. 43, 1994 10999
Letters in the neighborhood of the defect, and in this process, the local conversion of even a part of the strain energy to kinetic energy would give rise to appreciable local heating. The strain energy in the defect region and that from mechanical compression may thus be considered as a source that provides the energy for heating. In Figure 2b, the first hot spot at t = 1.7 was such a result: E h was only 0.069 at t = 0.8; but as the surrounding molecules relaxed toward the vacancy under load, E h at the hot spot was 0.524 at t = 1.7. In this case, the structural relaxation proved to be anelastic: When the crystal was reexpanded to its original volume, at the same rate as compression, beginning at t = 2.6, say, well after the first hot spot had occurred, the lattice reverted quickly to its original structure. Between t = 3 and 8, a number of other hot spots at different parts of the lattice also were generated by structural relaxation. But here the mode of relaxation was different and more widespread. It was accompanied by a decrease in Epsof about 0.15, and a corresponding increase in Ek,, indicating that plastic deformation had occurred in this time interval. The relaxation started as a kind of “buckling” which eventually developed into “kinks” in two (111) planes, in which rows of deformed molecules in successive pairs of (001) planes slipped part of a repeat distance in the [OlO] direction, the easy slip direction for this model. Both slip planes passed through the vacancy at the center, in adjacent periodic blocks of the crystal, where the ) especially slip had originated. Two peaks of E h ( ~ 1 . 1 5were prominent. The first occurred at t = 5 in the neighborhood of the vacancy. The second occurred at t = 6.9 when two “kinks”, started at defect sites in neighboring periodic blocks, met at the edge of the periodic block shown in Figure la. These “kinks” make up a slip system which may be considered as the counterpart of the shear bands that are widely observed in experiments. The details discussed here are, of course, the results of the specific initial and the boundary conditions imposed on this MD model. The relaxation process is expected to be affected by such factors as the lattice structure, the geometry of the molecules and of the defects, the direction of compression, etc. But the mechanism of hot spot heating, in all the cases investigated thus far, appear to remain the same; Le., the hot spot is generated by the local relaxation of a part of the strain energy, from the defects and from mechanical compression, and the conversion of this energy to kinetic energy and heat. In the present case, the hot spots caused by the “kinks” were both hotter and larger in size than the first hot spot around the vacancy defect. In Figure 2, the structure was in quasi-equilibrium after t = 14, but without reaching a hydrostatic state of compression. For t > 14, E h was again about 2 times &. In contrast, the hot spots from the vacancy and the “kinks” were typically 5-8 times the E k s of the system. Compression in Different Crystalline Orientations. The elastic constants of a cubic crystal usually change with axis rotation: For example, C ’11 of the present fcc monatomic crystal increased by 34% from the [1001 direction to the [ 1101 direction. Thus, in 1D elastic compression of the model to the same volume, AE,,, the energy stored in the lattice, as well as A&, and the rise in the stress components all would be higher in [llO] compression than in [loo] compression. And when the point of instability was reached and structural relaxation occurred, the energy released was also higher, and hence the hot spots were hotter, in [110] compression than in [loo] compression.
Discussion Fayer et aL5 and Fried and Ruggiero6 proposed that an enhanced rate of energy transfer under compression between intermolecular phonons and intramolecular vibrations is a key feature of hot spot generation, particularly at defect sites.5 In the present MD model, an enhanced intermolecular coupling indeed occurred under compression, especially in the neighborhood of defects. There was no arbitrary barrier to intermolecular energy transfer, except that produced by the impedance mismatch between molecules due to the different intra- and intermolecular interactions, and there was good thermal contact among the molecules. Moreover, there was sufficient energy source (AE,,), even at low compression, for hot spot heating, if any should occur. But significant hot spot temperatures were observed only when there was structural relaxation. In Figure 2b, the compression had started at t = 0, but the heating of the first hot spot in the neighborhood of the vacancy did not begin until t = 1, and this was accompanied by the onset of structural relaxation. Similarly, in other cases under elastic compression, when structural relaxation was absent, no hot spot developed of any consequence, apparently because the high-frequency phonons from the compression were not sufficiently energetic. This lack of “~p-pumping”~,~ could be due to the special method of compression employed here. Further investigation is needed to clarify this point. Dick7 has conducted experiments on the sensitivity of pentaerythritol tetranitrate (PETN) crystals to the direction of shock compression along different crystal axes. At low stress levels, PETN was found to be sensitive to compression in the [ 1101 direction and insensitive in the [ 1001 direction. Dick7 and Ritchie8 have discussed the results from the viewpoint of hindered shear in a bcc model of PETN. Despite the simplification of the fcc MD model employed here, results similar to those described by Dick were obtained. The expectation is that, in a simple 3D bcc model under comparable conditions, the hot spot also would be hotter in the [110] direction than in the [loo] direction. Now from the reference stiffness constants reported for PETN,9 the Young’s modulus, E, is actually found to be lower in the [110] direction than in the [loo] direction, while the reverse is true for the shear modulus, G. Dick’s results, however, showed that under similar compression the elastic longitudinal stress was almost 3 times higher in the [110] direction than in the [loo] direction, indicating that the shear strength of PETN was higher in [110] compression. Dick attributes the delayed onset of plasticity to hindered shear. In the context of the present study, the higher longitudinal stress would indicate a higher AEps (energy source) at onset of structural relaxation, and hence the hot spots would be hotter. Finally, dislocations in the molecular structures of energetic materials provide the possibility of acting as “in-situ’’ hot spots1° in a more significant manner than that described here for a single vacancy. This is because of the large open or hollow cores that are predicted to reside at the dislocation centers, due to the relatively low surface energy for these materials as compared with the larger strain energy centered at the dislocation position. Application of rapid compression or shock deformation to the dislocation core as a hole of, say, 5-10 molecular diameters would promote structural relaxation through individual molecules popping into the strain-free hole and thereby produce hotter and larger hot spots than would a single vacancy, but without significant movement of the dislocation.
Conclusions On the basis of MD model calculations, it is concluded that hot spots are generated from the combined action of (1) an
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increased strain energy at defects coupled with mechanical compression, thus creating a source of energy that is large compared with the kinetic energy of the system, and (2) local initiation of structural relaxation at defect sites, leading to the release and conversion of a part of the strain energy source to kinetic energy and local heating. The relaxation process also could generate additional defects such as “kinks” and slip planes, etc., which could move to other parts of the structure, thus enlarging the hot spot size and adding to the heating. The amount of heating at the onset of relaxation depends on the anisotropic strain energy stored in the crystal and the amount of energy released. In the present MD model, at low compression, the hot spot temperature was typically 5 -8 times higher than E b and about 3-4 times higher than E h from the Maxwell-Boltzmann distribution. Under comparable compression of either the fcc or the bcc lattice, the hot spots were hotter in the stiffer [ 1101 direction than in the [ 1001 direction. These results are consistent with experiments.
Acknowledgment. The authors express appreciation to R. S. Miller, Office of Naval Research, for giving encouragement for this work. Note Added in Proof. Additional data obtained with this model over a wider range (0-16%) of compression has been
reported in the paper entitled “Molecular Dynamics Modeling of Hot Spots in Monatomic and Molecular Crystals under Rapid Compression in Different Crystal Directions.” The paper was presented at the International Conference on Shock Waves and Condensed Matter, St. Petersburg, Russia, July 18-22, 1994, and will be published as a part of the conference proceedings.
References and Notes (1) Liebenberg, D. H., Armstrong, R. W., Gilman, J. J., Eds. Structure and Properties of Energetic Materials; Materials Research Society Sym-
posium Proceedings, Vol. 296; Materials Research Society: Pittsburgh, 1993. (2) Tsai, D. H. In ref 1, p 113. (3) Bandak, F. A.; Tsai,D. H.; Armstrong, R. W.; Douglas, A. S.Phys. Rev. B 1993, 47, 11681. (4) Tsai, D. H. J. Chem. Phys. 1991, 95, 4797. (5) Fayer, M. D.; Tokmakoff, A.; Dlott, D. D. In ref 1, p 379. Tokmakoff, A.; Fayer, M. D.; Dlott, D. D. J . Phys. Chem. 1993, 97, 1901. (6) Fried, L. E.; Ruggiero, A. J. In ref 1, p 35. (7) Dick, J. J. In ref 1. p 75 and references cited therein. (8) Ritchie, J. P. In ref 1, p 99. (9) Moms, C. E. In Proceedings of the Sixth Symposium (Internutional) on Detonation; ACR-221; Office of Naval Research, Department of the Navy: Washington, DC, 1976 p 398. The authors thank reviewer 2 for calling their attention to this reference and for commenting on the stiffness properties of PETN. (10) Armstrong, R. W. Revue Scientifique et Technique de la Dkfense 1994, 16, 161.
