A detonation wave profile is a particle velocity history that has been recorded once steady detonation has been achieved in an explosive sample. High speed diagnostics are required to resolve the detonation wave profile in experiments, and so VISAR (Velocity Inter- ferometer System for Any Reflector) or PDV (Photon Doppler Velocimetry) diagnostics are used at the interface between the sample and a window material [124]. Figure 4.5 shows the experimental configuration and a sketch of the resulting particle velocity histo- ries. The arrival of the detonation shock wave at the interface causes the particle velocity to jump up to a velocity related to the von Neumann spike state. Chemical reaction in the explosive causes the particle velocity to fall between the spike and the sonic point which, in a one-dimensional experiment, corresponds to the end of the reaction zone. In the laboratory frame, the flow is super-sonic until the sonic point and is not influenced by the depth of the explosive sample. Therefore, detonation wave profiles from different experiments or measured at different depths within a single experiment are expected to agree between the von Neumann spike and the sonic point, provided the same explosive and window materials have been used and steady detonation has been achieved. After the sonic point, the flow is sub-sonic and so the particle velocity traces from different experiments will differ, as illustrated by the multiple Taylor wave curves in figure4.5.
Explosive sample Taylor wave Interface velocity Time Window or PDV VISAR
von Neumann spike Sonic point
Figure 4.5: Sketch of the experimental configuration for detonation wave profile mea- surements (left) and the resulting particle velocity histories (right). The wave profiles from different experiments are expected to differ only after the sonic point when, in a one-dimensional experiment, the chemical reactions are complete.
Detonation wave profiles have been measured experimentally for HMX-based explo- sives [e.g., 124 and 184], but not for single crystals as far as I am aware. These data provide a useful test of the HMX model because detonation wave profiles are believed to depend on reactions that are due to bulk heating, rather than on the hotspot-driven reaction that is important in shock initiation [83]. Three features can be related to properties of the material model at the high shock strengths associated with detonation: the magnitudes of the von Neumann spike and sonic point are dependent on the unreacted and reaction prod- ucts equations of state, and the time delay between the spike and the sonic point is caused by the duration of chemical reaction. However, exact agreement between Peruse simu- lations of single-crystal HMX and the experimental data on HMX-based plastic-bonded explosives is not expected for two reasons. Firstly, after the sonic point, detonation wave profiles depend strongly on the experimental configuration. A model would need to be able to reproduce exactly the shock initiation behaviour in the experiment, and in par- ticular the run-to-detonation distance, in order to be able to match the detonation wave profile after the sonic point. This is not possible with a defect-free model whose run dis- tance is considerably longer than for a heterogeneous plastic-bonded explosive, owing to the absence of hotspots. Agreement can only be hoped for in the region between the von Neumann spike and the sonic point. Secondly, techniques are still under development to improve the quality of the detonation wave profile measurements, and current data are under-resolved for HMX-based explosives [184]. Until the experimental development work is complete, it would be premature to attempt a thorough assessment of the ability of the HMX model to represent it. Therefore, only a brief comparison will be made in this section to detonation wave profile data for PBX9501.
Figure 4.6: Simulated particle velocity histories for detonating HMX from Peruse and Menikoff[83], for comparison to experimental data for PBX9501 shot 1156 [124]. The particle velocity is recorded at the interface between the explosive sample and a PMMA window. Agreement between the simulations and the experiment is not expected beyond 25 or 30 ns after shock arrival.
methacrylate (PMMA) VISAR windows [124]. Interface velocity traces from one of his experiments, shot 1156, are given in figure4.6. The black and blue traces correspond to the two VISARs used in the experiments, with different velocity-per-fringe constants. The sonic point is estimated to correspond to an interface velocity of 2.89 km/s, which occurs between 25 and 30 ns after the shock arrival for this experiment. Shot 1156 used a Vistal gas-gun flyer to drive a flat-topped shock at 5.2 GPa into a 23 mm PBX9501 sam- ple, backed with a PMMA window. Since detonation wave profiles are insensitive to the experimental configuration until the sonic point, no attempt to reproduce the geometry of the experiments is made. The computational set-up comprises a 10 mm HMX sample backed by 1 mm PMMA, in plane geometry and with 200 zones/mm meshing. The HMX is initiated by adjusting the initial specific internal energy and density of the first 0.1 mm of HMX such that it is hot enough to react promptly at the start of the calculation. This is sufficient to initiate a detonation wave in the HMX, which propagates through the sam- ple to the PMMA window. The interface velocity is obtained by recording the particle velocity history of computational gauges in the first PMMA zone.
The detonation wave profile produced by the HMX material model in Peruse is com- pared to the experimental data from shot 1156 in figure4.6. The experimental data have been shifted in time so as to match the shock arrival time from the simulations. Pleasingly,
the simulated spike velocity is in good agreement with the experimental data. Although the situation is complicated by the presence of the PMMA window, this gives some reas- surance that the unreacted equation of state is accurate in this regime. Unfortunately, the following drop in particle velocity is not as fast in the simulations as in the experiments. This could in part be due to inaccuracies in the equation of state used for PMMA, but it is the contribution of the HMX material model that concerns us here. A series of Peruse simulations were used to show that the velocity drop is not greatly influenced by the unre- acted Hugoniot parameters, the specific heat of the unreacted HMX or the mesh density, but does depend on the reaction rate. In particular, increasing the Arrhenius parameter lnZ from 12.5 to 13, 14 or 15 (forZ in µs−1) causes the fall-off time to shorten signif- icantly. Comparing particle velocity histories from the Peruse simulation in figure 4.6 with burn fraction traces, reaction is 90 % completed within 4 ns of the shock arrival, and 99.9 % complete with 10 ns. Therefore, the sonic point occurs sooner in the simulations than it was estimated to do in the experiments. Agreement between the simulations and the experiments is not expected beyond the sonic point, or 1.115µs in figure4.6.
Menikoffhas also modelled detonation wave profiles in PBX9501 using an Arrhenius reaction scheme [83]. Figure4.6shows that Menikoffslightly over-predicts the spike but achieves a better match to the following drop in interface velocity, despite having a similar reaction duration to Peruse (reaction is quoted as being 90 % complete in 4.3 ns). This may be due to the different equations of state he used for both the unreacted explosive and the reaction products, which lead to a shock temperature of 2500K compared to∼3000K in Peruse. It should also be noted that Menikoffadjusted the reaction rate parameter lnZ
in order to match these experimental data. In section3.4, the reaction rate parameters for HMX were derived by using Hubbard and Johnson’s approximation to calculate explosion times consistent with Henson’s compilation of HMX data, neglecting the behaviour of the equations of state. It has already been demonstrated in section4.2that reaction rates are sensitive to the unreacted equation of state. Calculated detonation wave profiles are also sensitive to the reaction products equation of state, which defines the sonic point. In the future, the match to detonation wave profiles could be improved by combining equation of state improvements with adjustments to the reaction-rate coefficients.
In conclusion, figure 4.6 shows that the HMX material model gives a reasonable match to detonation wave profile data on PBX9501, validating the model in the detona- tion regime. Unfortunately, it has not been possible to validate the models for the binders in this way because wave profile data are not available.
4.4
Summary
The material models constructed in chapter 3 have been tested against available exper- imental data. The unreacted equations of state give a good match to Hugoniot data for HMX and the binder materials. Single-crystal Pop-plot data show that the HMX model brackets the data within the uncertainties on solid heat capacity, validating the reaction rate in the shock initiation regime. Detonation wave profiles for HMX compare reason- ably well with data for PBX9501, validating the model in the detonation regime.
Shock heating of crystals and binder
The simplest possible hotspot mechanism in plastic-bonded explosives is shock heating of HMX crystals and binder. Owing to their differing material properties, higher tem- peratures are produced in the binder than in the crystals when the explosive is shocked. Subsequent chemical reactions may heat the binder further, amplifying the temperature difference between crystal and binder regions. Although this is not generally regarded as a hotspot mechanism [17], it does cause significant temperature localisation owing to the position of the binder in pockets surrounding the crystals. Therefore, it is possible that heat conduction from the binder to the crystals could cause sufficient chemical reaction in the crystals for them to explode. The accumulating pressure and temperature caused by the reaction of multiple crystals could eventually lead to detonation in the explosive.
Since the dominant physics controlling this process is heat conduction and Arrhenius chemistry, early work investigating this possible hotspot mechanism used ReactDiff. Al- though the results in section5.1indicate that the shock heating of crystals and binder is not a feasible hotspot mechanism, the effect of hydrodynamics could not be discounted. Section5.2will explain how Peruse was used to account for the combined effects of hy- drodynamics, heat conduction and chemistry, and to determine conclusively whether this is a feasible mechanism in PBX9501 and EDC37. Since the material properties data for the HMX and binder components of these explosives are not well known, it is important to check that the uncertainties in the material properties do not affect the results. Ex- ploratory Peruse simulations to determine the influence of these modelling uncertainties on the results are presented in section5.3.
5.1
Initial investigation using ReactDiff
The following is a brief summary of an initial investigation into this hotspot mechanism. ReactDiff [115] was used to simulate spherical crystals of HMX with radius 0.45µm coated in a thick layer of binder. ReactDiffsolves the reactive-flow equations2.7neglect- ing hydrodynamics and species diffusion. McGuire & Tarver’s [86] three-step reaction scheme A →1 B →2 2C →3 D was used for the HMX (equations 2.15) and the binder was treated as inert. Illustrative results are given in figure5.1 for initial conditions rep- resentative of a shock at 9.5 GPa input pressure. The temperature profile shows that the HMX (with radius <0.45µm) is initially cooler than the binder (radius >0.45µm). For
∼1.5µs, heat conduction from the binder causes the temperature of the HMX to increase. The species histories show that the endothermic reaction A→ B progresses over the first 1.3µs, until the first exothermic reaction B→ 2C takes hold and causes the temperature in the centre of the HMX crystal to rise further. The third, highly-exothermic reaction 2C → D takes over at 1.85µs when there is a sudden increase in the HMX temperature and the composition transforms almost instantaneously to pure D. This sudden transfor- mation is a thermal explosion and the time at which it occurs depends on the size of the HMX crystals, and the thickness and reactivity of the binder layer.
For a 9.5 GPa input shock, the minimum time to explosion observed for any crystal size or binder thickness was 1.5µs for PBX9501 and 1µs for EDC37. These timescales are well above the run-to-detonation time of 0.35µs in PBX9501 and 0.55µs in EDC37, extrapolated from experimental Pop-plot data [12, 134]. This implies that, if shock heat- ing of crystals and binder were the only hotspot mechanism in simulations of these ex- plosives, none of the HMX crystals would explode within the experimental run-time and detonation would certainly not be produced. ReactDiffsimulations were also undertaken using initial temperatures representative of input pressures between 1.3 and 6.3 GPa. No thermal explosion occurred within 10µs, in contrast to experiments which detonate within 10µs at these shock strengths. These observations suggest that thermal explosion caused by shock heating of the binder alone is not responsible for shock initiation in PBX9501 and EDC37.
However, ReactDiffis a static code and does not solve the hydrodynamic equations of motion. The simulations of crystals and binder, described above, had significant tem- perature variation through the computational domain and the only mechanism acting to smooth this was heat conduction. In reality, as the temperature of the crystals and binder
Figure 5.1: Illustrative results for a ReactDiff simulation of a spherical HMX crystal surrounded by a layer of PBX9501’s binder. The graphs show HMX species histories (above) and temperature profiles (below). The time to explosion is∼1.85µs when there is a sudden increase in the HMX temperature and the composition transforms almost instantaneously to D.
rises, the pressure also increases. In addition, high-pressure gaseous reaction products are produced when the crystals and binder react. Waves will distribute this increase in pressure throughout the computational domain on timescales several orders of magnitude faster than heat conduction. It was concluded that shock heating of crystals and binder could not be eliminated as a hotspot mechanism until hydrodynamics had been accounted for.