MOLECULAR DYNAMICS SIMULATIONS OF THE PROPAGATION OF A DETONATION WAVE IN A CRYSTAL

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MOLECULAR DYNAMICS SIMULATIONS OF THE PROPAGATION OF A DETONATION WAVE IN A

CRYSTAL

M. Peyrard, S. Odiot, E. Oran

To cite this version:

M. Peyrard, S. Odiot, E. Oran. MOLECULAR DYNAMICS SIMULATIONS OF THE PROPAGA- TION OF A DETONATION WAVE IN A CRYSTAL. Journal de Physique Colloques, 1987, 48 (C4), pp.C4-291-C4-301. �10.1051/jphyscol:1987421�. �jpa-00226660�

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Colloque C4, suppl6ment a u n"9, Tome 48, septembre 1987

MOLECULAR DYNAMICS SIMULATIONS OF THE PROPAGATION OF A DETONATION WAVE IN A CRYSTAL

M. PEYRARD, S. ODIOT* and E . ORAN'*

Laboratoire O.R.C., Facutte des Sciences, 6 boulevard Gabriel, 21100 Di jon, France

*D.R.P. Universit6 Pierre et Marie Curie, Tour 22, 4 place Jussieu, 75252 Paris Cedex 05, France

* +

L.C.P. and F.D. Code 4410, Naval Research Laboratorg, Washington, D.C. 20375, U.S.A.

Nous presentons une approche microscopique des ondes de detonation induites par choc dans les solides qui utilise la dynamique moleculaire pour determiner la structure du front de dbtonation. Nous montrons que des resultats utiles peuvent &re obtenus h partir d'un simple modele uni-dimensionnnel puis nous presentons les resultats de simulations sur un modele bi- dimensionnel. Les echelles d'espace et de temps accessibles a la dynamique moleculaire sont discutees et nous decrivons la structure de I'onde de detonation. Nous montrons que la quantite d'energie transferee dans les modes transversaux joue un role fondamental pour determiner si un solide peut supporter des ondes de detonation et determiner leur vitesse quand elles existent. La simulation de systemes de plus grande taille montre que, dans son Btat stable, le front de detonation est oblique par rapport 2 la direction de propagation. Ceci suggbre que les structures courbes associees aux cellules de detonation pourraient avoir une analogie au niveau microscopique. Des extensions possibles de la methode sont presentees.

Abstract

We present a microscopic approach to shock induced detonations in solids which uses molecular dynamics to determine the structure of the detonation wavefront. We show that some useful results can be deduced from a simple one-dimensional model and then present two-dimensional simulations. The time and space scales that can be investigated in this approach are discussed and the structure of the detonation wave is described. We show that the amount of energy transferred into transverse modes plays a crucial role in determining if a solid can sustain a detonation and the speed of the detonation wave when it exists. Simulations of larger systems show that in the stable state the detonation front is oblique to the direction of the propagation.

This suggests that the curved structures at the detonation front associated with detonation cells may have an analogy at the microscopic level. Possible extensions of the method are discussed.

I. Molecular~dynamics : why ?

A detonation wave is a supersonic shock wave sustained by the energy released in exothermic chemical reactions. The energy exchange is complex because the shock provides energy for chemical dissociations and receives energy from the exothermic reactions. Since detonations are indeed macroscopic phenomena, they have been investigated primarily by macroscopic theories. Informations such as the detonation speed in the steady state can be deduced from conservation of mass, momentum and energy. In general the thermodynamical approach is successful in treating gas-phase and some liquid-phase explosives but the properties of condensed phases, especially solids, are not well understood. In addition, macroscopic theories cannot explain some recent experimental results on solids which are clearly related to the microscopic structure of these compounds : (i) strong correlations have been observed between detonation properties and the existence in the crystal of structures connected by strong bonds (I), (ii) a large anisotropy in detonation speed and sensitivity to shocks

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987421

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have been found in experiments on monocrystals (2) .

Consequently, progress in understanding detonations in solids requires investigations at the microscopic level. As it has been done for the macroscopic scale investigations, it would certainly be desirable to start the microscopic investigations by experiments before a satisfactory theory can be built. This would require observations at the molecular scale to determine, for example, the structure of the shock wave at the atomic level, how the atoms move and how their motions induce bond breaking. Unfortunately experiments at this small scale are extremely difficult. Attemps have been made to use high-resolution measurements to determine the width of the shock front, but they cannot attain a resolution of the order of the atomic spacing. Investigations of ultrasonic attenuation in solids give some indications of the energy exchange between acoustic waves and atomic vibrations that may lead to dissociations(3) but it would be most useful to observe atomic motions directly. Molecular dynamics offers an alternative : the n.-u

The basic idea of the molecular dynamics approach is to use Newton's law of motion to determine the behavior in time of coupled particles given the forces between them and initial and boundary conditions. The simultaneous motions of a great many particles can be calculated numerically using high-speed computers. We can observe the effect of a shock on the material, record atomic displacements, determine the structure of the shock front, etc. The resolution of the "experiment" is very high since we can, for instance, observe the motion of a single atom. But we cannot avoid an "uncertainty principle" : this high accuracy is not obtained on a real explosive but on a model, and the validity of the results may depend on the choice of that model. In addition, there are severe restrictions on the space and time scales that can be observed. One objective of this paper is to show that, in spite of its limitations that should not be ignored, molecular dynamics is an efficient tool for understanding detonations at the microscopic level.

Only after the microscopic observations have been performed, can we try to propose theories to explain them and make the connection w i h the macroscopic observations. This aspect is also very interesting and has motivated our efforts.

II. "Experimental" molecular dynamics results.

It is presently beyond our computing possibilities to try to model a real explosive which may have several tens of atoms in a unit cell. This would also be meaningless since the validity of a calculation is determined by weaker points: although we do know atomic positions to a high accuracy, we do not know atom-atom interaction forces to the same accuracy. Thus we cannot expect quantitative results from a molecular dynamics calculation.

Therefore we focus on obtaining a fundamental understanding of the phenomena using simple generic models that do not attempt to mimic a specific crystal.

j l One . . ₍₄₉₅₎

This is the simplest model that includes the features essential for modeling the propagation of a detonation wave. More detailed results are described later for two dimensional models. Here we give a short account of the results of this simple case to illustrate how useful results can be derived from a very simple model.

Fig.1 : Schematic drawing of the one dimensional model consisting of a chain of "diatomic molecules". The atomic displacements are restricted to the direction of the chain.

The model is shown schematically in Fig.1. In a one-dimensional array of two components N-C molecules, only displacements along the direction of the chain are allowed. In a detonation wave these displacements can be as large as half the cell size so that the nonlinear character of the bonds must not be ignored. For this purpose, the N-N or C-C coupling are described by Morse potentials. "Realistic" values of the potential parameters have been obtained from atom-atom pair potentials known for molecular crystals. The intramolecular potential is a predissociative exothermic potential, obtained by substracting two Morse potentials with appropriate parameters, to reproduce the potential energy barrier and dissociation energy determined for the nitromethane molecule from ab initio calculations(6) . When a strong impulse is applied to the first N-C molecule in the chain and the N and C sublattice are different (with different bond strengths or masses), the N and C displacements may be different enough to extend the N-C distance to its breaking point. The N-C interaction is now repulsive and this repulsion can promote the propagation of a compressive shock in the lattice which breaks the next bonds. The detonation wave is the transmission of this bond breaking wave along the lattice. Each time a molecule is dissociated, some potential energy is converted into kinetic energy of the atoms. Since the one dimensional model cannot describe transverse energy transfers and other energy-loss mechanisms, we introduced phenomenological damping terms in the equations . The damping coefficient is the only adjustable parameter that cannot be estimated apriori in the model.

What can we learn from such a simplified model ?

(i) First the model reproduces the main features of a detonation wave in a solid. In particular we observe a shock front separated from a reaction zone by some induction zone in which the shock front has already passed but the N-C bonds are not yet broken. The detonation propagates at a velocity characteristic of the model and not of the initial excitation. Contrary to what is generally found for one-dimensional simulations of shock-wave propagation in solids, the profile of the detonation wave is steady. There are two different thresholds for the excitation impulse:

a dissociation threshold over which some molecules are broken but the detonation finally dies out, and a larger detonation threshold which is the minimum impulse required to initiate a steady detonation.

(ii) More important, this model helps us to understand some of the experimental results mentionned in Section I. A detonation can propagate in the chain only when one of the two sublattices is sufficiently rigid. The role of this rigid sublattice is to insure a sufficient coherence in the propagation. On the contrary the second sublattice must be "soft" to allow the large atomic displacements that are necessary for the bonds to break. These observations are in agreement with the correlations found between detonation properties and crystal structure (1).

However the model is oversimplified and does not allow investigations of the role of the crystalline geometry. In addition, the phenomenological damping parameter cannot be estimated satisfactorily. This prompted us to introduce additional degrees of freedom by investigating two-dimensional models.

7 ) T w o - d i m e n m ( 7 . 8 )

Except in certain specific layer compounds, the "real world" is three-dimensional. However the computing effort increases as a power law of the dimensionality of the model, so that three-dimensional calculations are very expensive or must be restricted to small systems. While the transition from one dimension to two dimensions qualitatively changes the results by allowing transverse motions, we do not expect the transition from 2D to 3D to be so important. This is the reason for our choice of a two dimensional model.

a)The model. Crystal structure and bonding.

The structure of the model lattice, shown schematically in Fig.2, consists of an array of molecules made of two- components N and C which can represent more complicated groups that occur in real crystals. For instance, N and C could represent NO2 and CH3, respectively. Following the results obtained with the one-dimensional model, the two sublattices N and C are chosen to be different. We denote by N the more rigid sublattice, i.e., the lattice

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with the stronger chemical bonds or the more massive components. The X axis is the axis along which the detonation propagates. The parameters that characterize the lattice are the dimensions a and b of the unit cell along the X and Y axes, the equilibrium length dNC of the N-C bond and the angle a between the N-C molecule and the X axis. The bonding scheme is similar to the one used in the one-dimensional case. The intramolecular N- C potential is the same predissociative potential generated by the difference between two Morse potentials and the intermolecular potentials are Morse potentials. Each atom is connected by intermolecular bonds to its nearest neighbors and next nearest neighbors.

Y

j + l

j Fig. 2. Schematic picture of the two-

dimensional model.

j - 1

Since the calculation has been designed to simulate a solid phase in which the neighbors of a given atom are predetermined, we do not need an algoritm to find these neighbors at any time. This saves computing time and allows us to take full avantage of the known structure to vectorize the program for efficient computation on the Cray. However, since some atoms, especially the C atoms which belong to the "soft" sublattice, may move rather far from their original neighbors, we had to introduce some long-range interactions. These are not important in the initial state but start to play a role when dissociations occur. The interaction potentials are written so that the lengths of the bonds in the crystal structure are minima of the potential when the lattice is at equilibrium.

Consequently we can impose a given configuration on the lattice. Because transverse motions are now part of the calculation, we have not included any phenomenological damping term in this model. Therefore there is no parameter in the model that cannot be estimated a priori .

b) Molecular dynamics techniques.

Although it is not our aim here to describe the details of the calculations, it is useful to give an idea of the method of solution since it determines what can be deduced from molecular dynamics calculations. We solve the classical equations of motion of the atoms moving in prescribed potentials. In order to reduce the number of equations, we restrict the study to atoms situated in a window that moves along the lattice together with the detonation front.

This allows us to investigate the propagation of the detonation along several hundred cells even though the window includes only 40 to 256 cells in the X direction. Even with this technique

are of the order of 1 micrQn. The initial impulse that initiates the detonation is applied at the left end of the lattice (index i=l) during a limited time t i and then this end is left free.

Along the Y direction, the lattice has only 4 to 20 cells and we use periodic boundary conditions in this transverse direction. Time is divided into time steps At and the algoritms used to solve the equations determine the state of the system (atomic positions and velocities) at time t+At as a function of the state at time t. They are accurate only

when At is much smaller that the caracteristic times of any motion in the system, i.e., in this case the periods of atomic motions in a solid. This imposes At = 10-16 or 10-15 s. Since simulations can now hardly exceed 106 time

steps, -hat can be inv-amics is at verv most a nanos- Consequently

both the space and time scales of molecular dynamics simulations are extremely small compared to a macroscopic experiment.

Direction of the detonation wave

t ^tot,

First dissociations:

Reaction front

Fig.3 : Typical aspect of the observation window showing the three main regions : unreacted region, induction zone and reaction zone.

cf Results on a "smaii" system

Time

FigA : position of the shock front (solid line) and of the last bond broken (crosses) as a function of time when a detonation propagates in the two dimensional lattice at zero initial temperature.

In a typical experiment the observation window is divided into three regions, as shown in Fig 3.

The material ahead of the shock front is still in its initial state. It can be either at rest (simulation with zero initial temperature) or the atoms can have random thermal motions. We call reaction front the region where the first dissociations occur. It is separated from the shock front by an induction zone. In the reaction zone behind it, most of the intramolecular N-C bond are broken.

Let us consider first the results obtained in systems with 4 to 8 cells in the transverse (Y) direction. Figures 4 and 5 present typical results for a detonation propagating in a lattice initially at rest. Figure 4 shows the position of the shock front and the location of the most recently broken N-C molecule as a function of time. Note that dissociation follows the shock frontwith very little delay. This figure also shows that there are two propagation regimes with different propagation speeds. The high-speed regime, immediately after the initiation corresponds to the propagation of almost purely longitudinal motions. A compressive wave in the N sublattice and a rarefactive wave in the C sublattice stretch the N-C bonds. When the stretching is large enough, the N-C interaction becomes repulsive, pushing the C atoms backward and the N atoms foward.

This causes both the rarefaction wave in the C sublattice and the compression wave in the N sublattice. The atomic motions in this regime are very regular and the detonation propagates in the X direction as a perfectly plane

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wave. The shock front profile is not steady because the compressive wave in the N sublattice tends to separate into localized solitonlike excitations. Since the 2D model does not include a dissipation term and because the energy transferred into transverse motions is very small, almost all the energy of the initial impulse stays in the detonation wave front. The result is that, in this first propagation regime the detonation speed depends on the energy of the initialshock. This result is contrary to both the experimental results and the results of macroscopic fluid models of detonations. The conclusion is that this regime does not exist in real systems. Our calculations show that this regime is notstable. After some time which depends on the amplitude of the initial shock, it evolves spontaneously into a lower speed regime causing the "knee" in Fig.4. The structure of the detonation wave which propagates in the lower-speed regime is shown in Fig.5.

The lower-speed regime is much less regular than the high-speed state. Longitudinal and transverse motions exist with wavelengths of a few unit cells, and the detonation no fonger propagates as a perfect plane wave. However, it is important to notice that coherent propagation persists. The detonation still consists of a compression wave in the N sublattice and a rarefaction wave in the C sublattice. The shock frontpropagates now with a permanent profile ,i.e., its width stays constant and, although it fluctuates slightly, the distance between the shock and reaction fronts keeps a constant mean value. A significant part of the energy released in the dissociation of the N-C molecules is transferred into transverse oscillations that do not contribute to motions in the direction of the propagation. As a result the speed of the detonation wave is much smaller than in the first regime

. . .

~t IS lnd- of the initial e x c m It is a characteristic of the material which results from the way the energy released in the chemical dissociations is partitioned between transverse and longitudinal

(a) N atoms (a) C atoms

(b) N atoms (b) C atoms

Fig.5 : Atomic displacements when the detonation propagates in the 2D lattice at the lower-speed regime. (a) Longitudinal and (b) transverse displacements of the N and C atoms shown in the vertical direction on a lattice grid.

motions. This lower-speed regime persits when the simulations are performed with a lattice initially in random thermal motion. In that case, an abrupt transition from the initial high speed regime to the lower speed regime is not observed because the perfectly coherent high speed regime does not exist. The detonation gradually slows down to the speed characteristic of the material which is smaller than when the initial temperature is zero because the random fluctuations tend to increase the role of the transverse motions. For both zero or non-zero initial temperature, the shock front includes only a few cells, so that it is sharp on an atomic scale and thermal equilibrium is not reached until long after the shock front passes. This raises important questions about the validity of any macroscopic thermodynamical theory which describes the shock region with an equilibrium equation of state.

As for the I D model, there are two threshold values of the initial impulse, the dissociation threshold and the detonation threshold which is larger. The model also gives a number of,microscopic conditions that a crystal must fulfil in order to sustain a detonation wave. The need for two unequivalent sublattices, that was seen in the 1D case is confirmed. However the 2D model also shows that the crystal structure plays a crucial role because it determines the amount of energy that is transferred into transverse motions. The lattice parameters a and b determine the detonation speed. The larger the alb ratio, the greater the detonation speed. This can be understood from a geometrical analysis of the forces in the lattice because large ah ratios increase the weight of the longitudinal components. Moreover we found that when a/b is too small ( a/b < 1.2 ), a steady detonation cannot be generated whatever the amplitude of the initial impulse. The positions of the atoms in a unit cell (parameter a in our model) has little effect on the detonation speed, but determines the sensitivity to shocks. As for the 1 D case we find that detonation speed and sensitivity to shock are related to different microscopic parameters which may explain why they are not strongly correlated in experiments.

d) Extensions, investigations of larger systems.

Figure 6 shows the results of a numerical simulation similar to those presented above, but performed in a lattice with 20 cells in the transverse Y direction instead of 4 or 8. The two propagation regimes (high speed and lower- speed) are still observed, but the lower-speed regime is no longer stable. A small velocity decrease suggesting a third regime can be observed after the detonation has moved along about 200 cells and, shorty after, the propagation of the dissociations stops. This effect is not due to having an initial impulse below the detonation threshold because increasing the impulse does not prevent the detonation from dying. This

2 4 7 indicates that the conditions for a solid to sustain

a detonation wave deduced from simulations on small systems, although they are necessary

-

x observed, are not sufficient. This poses three

u C 0

- questions : (i) why does a detonation propagates

- -

6 in a small system and not in a large one ? (ii) what

physical process is missing in the model that may prevent large systems to sustain a stable detonation ? and (iii) what is exactly the nature of this third regime that appears before the

o detonation stops ? The first question is answered

0

Time 5 0 0

by noting that, in small systems, the periodic Fig.6: Position of the shock front (solid line) and of the last bounda~ in the transverse bond broken (crosses) as a function of time for a simulation have an effect which is equivalent to a strong in a lattice with 20 cells in the transverse direction. Note the

change of slope sl~ghtly before the dissociations cease COnfineemnt. This effect is much

propagating. smaller in large systems.

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The second question is answered by noting that, in real systems, an energy release occurs in the reaction zone while this is not the case in our model. In order to model the exothermal reactions between the fragments after the dissociation of the N-C molecules, the energy that these reactions release has been introduced in the calculations. For this purpose the kinetic energy of the fragments is increased with a rate Q that depends on the reactions kinetics. Since we expect the exothermal reactions to start after a finite delay following the N-C dissociation, the energy of the fragments is increased only when the distance between two fragments of a given molecule exceeds k.do where do is the N-C distance for which the intramolecular interaction becomes repulsive and k a constant which depends on the delay (generally fixed to k=4). Even a value of Q that is rather small compared to the energy released in the dissociations is sufficient to stabilize the third propagation regime, and so a detonation can again propagates permanently.

Once the third regime is stable we can investigate its nature and answer the third question. Figure 7 shows a typical aspect of the lattice in this third regime. Figure 7a is a map of the lattice showing the shape of the unreacted region, induction zone and reaction zone. The longitudinal displacements of the N atoms are plotted in Fig 7b. The interesting features in Fig. 7 are the oblique patterns.

Fig .. 7 : Propagation of a detonation in the third regime in a lattice with 20 cells in the transverse direction .

(a) map of the lattice showing the regions that have not yat been reached by the shock ( white rectangles)

.

The detonation no longer propagates as a plane wave perpendicular to the direction of the propagation (X axis) but forms now with it an angle which is of the order of 45" for the crystal structure investigated in this calculation.

This allows even more energy to flow into transverse modes and explains why the detonation dies if the energy released behind the detonation front is not properly taken into account. Similar patterns are always found after the third regime is established and we have not found any indication of an instability of this regime. The oblique propagation could not be observed in simulations of smaller systems because the periodic boundary conditions imposed in the Y direction forced the shock and reaction fronts to stay perpendicular to the X axis. This points out the importance of the details of the model (and boundary conditions) in the numerical experiments and stresses the caution that must be taken in interpreting the results.

It is tempting to connect the oblique patterns observed in the third regime to the observation of detonation cells which are well known in gas detonations but exist also in solids(8). Since molecular dynamics investigate a domain very much smaller than the size of a detonation cell, we do not expect to observe a full cell pattern. However cells are formed in experiments because the propagation of a planar detonation perpendicular to the propagation of the detonation is unstable. This is similar to what we find at the microscopic level. In addition cells are generally not observed when the detonation fails. This is also what we find at the microscopic level.

When the initial impulse is below the detonation threshold or when Q is too small for a stable detonation to propagate, the shock front stays perpendicular to the X axis. This results strongly suggests that the cellular patterns found in solid detonations may well have a microscopic origin. If this is the case the patterns would certainly be very sensitive to crystal structure which is consistent with the observation of large variations in cell size between cast and pressed TNT(9). However, our observations of the third regime in the microscopic investigations of the propagation of a detonation are very recent and further investigations are necessary before conciusions can be drawn.

Ill. Discussion. Further developments.

The main conclusions that result from our numerical experiments are :

(i) At the microscopic level in a solid, a detonation wave propagates as a coherent excitation which has a complex structure. The shock front maintains its shape as it propagates and is narrow on a molecular scale. Thermal equilibrium is not achieved until long after the shock passes.

(ii) In order to sustain a detonation, a crystal must have a rigid sublattice, which is necessary to insure the coherence of the propagation.

(iii) The crystal structure determines the amount of energy transferred into transverse modes. The detonation speed is mainly determined by the balance between the energy released in the chemical reactions and the energy that flows into transverse motions. When the latter is too large, the detonation fails.

(iv) At the microscopic scale, a planar detonation front perpendicular to the direction of the propagation of the detonation is not stable. In the stable state, the front forms an angle with the direction of the propagation. This suggests that the detonation cells observed in solids may well have a microscopic origin.

Although one should be cautious in extending the results to a given real material due to their large sensitivity to the details of the crystal structure, they are supported by experimental evidence. We have already mentionned the correfations found between detonation properties and strongly bonded components in crystals.

In addition, liquid nitromethane is a well known explosive although, to our knowledge, all attempts to detonate it as a crystal failed. This could be understood in the framework of our results. The crystal structure shows some zig- zag chains so that strong compression of the solid results in very large energy transfer into transverse modes which could cause the detonation failure. Finally very large anisotropies in shock-initiation sensitivity of

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monocrystals of PETN have been reported(2). The interpretation that was proposed for these results is that the directions in which the detonation is not initiated is the direction in which the initial shock causes the largest transverse motions. This is consistent with our results.

if 13 Nssible to make many improvements in these calculations. For example, in our model, the prediCcki8tive exothermic state is assumed to be the initial state of the N-C molecule. This means that this study is strictly valid only for primary explosives where such a ground state has been found. For secondary explosives where the predissociative state is an excited state, the transition to this excited state should be included in the model. It would also be very interesting to treat the reaction zone better in the computations because important phenomena occur in this region. It is not meaningful to extend the calculations to model realistic crystal structures as long as the atom-atom forces are not better known. It would be more interesting to introduce some quantum effects to obtain a better description of the energy exchange between the external vibrational modes involved in the shock and the intramolecular electronic states.

Finally, the real challenge is to find a way to bridge the gap between the microscopic and macroscopic approaches. One possible approach is to use theories of nonlinear excitations in solids (and particularly solitons) to obtain a description of the detonation wave. This approach has given interesting results for one-dimensional models of shock waves in lattices. However, in two or three-dimensional systems, and due to the complex interplay between chemistry and mechanics in energetic materials, our simulations show that simple solitons do not exist. This makes the analytical treatment of the detonation wave more difficult. Another possible approach that has already yielded interesting results in hydrodynamics might be to use cellular automata simulations and determine what should be the properties of an individual cell to reproduce the macroscopic features found in experiments. Then the comparison with the molecular dynamics simulation could connect these properties to microscopic quantities. Such a process, scanning along the different scales, would be similar to the renormalization group techniques that have been so fruitful in phase transitions investigations(l0). However, it might not be able to produce quantitative results.

Aknowledaements.

The work presented here have been partly supported by the DRET and the CEA through the delegation of M.P.

in its "DBpartement de Physico-Chimie". The computations have been performed with the Cray XMP of'the Naval Research Laboratory in Washington and with the Cray 1 and Cray 2 of the T e n t r e de Calcul Vectoriel pour la Recherchen through a research grant with the =Departement de Recherches Physiques de l'universit.3 Pierre et Marie Curie".

References.

1 - G. Bertrand and J. C. Mutin (unpublished) 2 - J. Dick, Appl. Phys. Lett. ,44,859 (1 984) and

H. W. Koch and Ch. Baras , Rapport 28/71. lnstitut Franco Allemand de recherches de Saint Louis, France, 1971 (unpublished).

3 - A. Zarembowitch, paper presented in this conference and references therein.

4 - M. Peyrard, S. Odiot, E. Lavenir, C.R. Acad. Sci. 299, Sdrie 11, 917 (1984) 5 - M. Peyrard, S. Odiot, E. Lavenir, J. Schnur, J. Appl. Phys. 57,2626 (1985) 6 - C. Mijoule, S. Odiot, S. Fliszar, J.M. Schnur , Theochem. 149.31 1 (1987) 7 - M. Peyrard, S. Odiot, E. Oran, J. Boris, J. Schnur, Phys. Rev. B 33,2350 (1986)

8- S. Odiot, M. Peyrard, C. Mijoule, J. Schnur, E. Oran, Int. J. of Quantum Chemistry XXIX, 1625 (1986) 9 - P. Howe, R. Frey, G. Melani, Combustion Science and Technology, 14,63 (1976)

10 - J. Lee, private communication.

Q u e s t i o n s

-

1) Why i s a damping t e r m necessary t o o b t a i n a s t a b l e d e t o n a t i o n when you have gone t o such t r o u b l e t o o b t a i n "reasonable" p o t e n t i a l s ?

2) How i s i t t h a t y o u r e x p l o s i v e propagates when a r e a l e x p l o s i v e has a f a i l u r e d i a m e t e r >> t h a n t h e d i a m e t e r s y o u have s t u d i e d ?

3 ) Do y o u t h i n k t h a t y o u r c o n c l u s i o n s a b o u t t h e importance o f t r a n s v e r s e vs l o n g i t u d i n a l waves and t h e s l i p p a g e o f " s o f t " c h a i n s o a s t "hard" c h a i n s a r e c l o s e l y r e l a t e d t o t h e f a c t t h a t y o u r e x o t h e r m i c bond b r e a k i n g ~ o t e n t i a l i s p l a c e d p a r a l l e l t o t h e shock f r o n t . I n o t h e r words, a r e y o u r r e s u l t s i m p l i - c i t l y t i e d t o t h e o r i e n t a t i o n o f y o u r energy r e l e a s i n g f u n c t i o n ?

RGponse -

1 ) As Don TSAI e x p l a i n e d , even i f we d o n ' t p r e t e n d t o mimic r e a l systems, we must pay a t t e n t i o n n o t t o work on a model t h a t p r o v i d e s unreasonable r e s u l t s .

I n t h e 1 D case many c h a r a c t e r s o f t h e r e a l 3 D w o r l d a r e absent. We cannot t r e a t them i n t h e framework o f t h i s s i m p l e case b u t we cannot s i m p l y i g n o r e them. The t e r m i s t h u s a k i n d o f compromise. The " r e a l i s t i c " p o t e n t i a l s were e s s e n t i a l l y h e r e t o have c o r r e c t r a t i o s between t h e N02-NO2 bond s t r e n g t h and CH3-CH3 bond s t r e n g t h .

2) I n a M D c a l c u l a t i o n ? p e r i o d i c boundary c o n d i t i o n s a r e a way t o s i m u l a t e (as w e l l as we can) an i n f i n i t e system. Thus i n f a c t t h e system i s n o t so s m a l l and i n p a r t i c u l a r t h e energy cannot escape t r a n s v e r s a l l y . The complete system i s c o n s e r v a t i v e . What t h e s t r u c t u r e changes i s t h e r a t i o between l o n g i t u d i n a l and t r a n s v e r s e energy n o t t h e t o t a l amount o f m g y i n t h e sys tem.

3 ) I d o n ' t have y e t f u l l r e s u l t s f o r wide systems. I n s m a l l e r systems (Ny q 8) t h e o r i e n t a t i o n o f t h e N-C bond ( e x o t h e r m i c bond) has l i t t l e e f f e c t on t h e genera? s t r u c t u r e o f t h e d e t o n a t i o n wave as I see i t . I t m o d i f i e s i t on a s m a l l s c a l e b u t I s t i l l observe r o u g h l y a compressive wave i n t h e r i g i d s u b l a t t i c e and a r a r e f a c t i v e one i n t h e s o f t s u b l a t t i c e .

My f e e l i n g f o r l a r g e systems i s t h a t t h e o r i e n t a t i o n o f t h e N-C bond w i l l a f f e c t t h e tendency f o r t h e f r o n t t o become o b l i q u e ( o r p o s s i b l y t h e a n g l e between t h e f r o n t and t h e d i r e c t i o n o f t h e p r o p a g a t i o n ) b u t t h i s has t o be checked.

C o m m e n t a i r e - I?. MAUSOU

Je pense q u ' i l y a l i e u d ' e x a m i n e r assez rapidement ( a f i n d ' e v i t e r des ma- l e n t e n d u s l o r s des c o n f r o n t a t i o n s u l t e r i e u r e s de l a t h e o r i e fondee s u r l a dynamique m o l 6 c u l a i r e , avec c e l l e c l a s s i q u e ) c e que s o n t dans l e s deux cas

" l ' o n d e de choc" e t "7'onde de d e t o n a t i o n " : I 1 me semble en e f f e t que dans v o t r e approche l ' o n d e de choc n ' e s t pas d e f i n i e , comme dans l a t h e o r i e c l a s s i q u e , Dar r e f e r e n c e a l ' o n d e sonore c ' e s t a d i r e 8 une ~ e r t u r b a t i o n r e v e r s i b l e ( a u sens thermodynamique) d o n t l a c e l e r i t e e s t d e f i n i e p a r l a c o m p r e s s i b i l i t e du m i l i e u .