Arrhenius chemistry parameters

In section3.4, no maximum and minimum values were quoted for the heat of reactionq1 owing to the difficulty of estimating the uncertainty on q1, since it is necessary that the heat of reaction from the chemistry scheme be consistent with the JWL reaction products equation of state. The effect of q1 on the simulations is likely to be two-fold. It will influence the temperature reached in the detonation products, rather like the changes to

TCJandcv,CJdid. It will also influence the time to explosion. Using Hubbard & Johnson’s approximation for the time to explosion from a single step Arrhenius rate

t= cv,sT 2_R Z1q1E1 exp _E 1 RT ,

Figure 5.19: Temperature histories for alternating HMX and binder calculations of EDC37, withTCJandcv,CJvalues for the binder adjusted according to the numbered list in the text, to investigate the effect of uncertainties in the thermal properties of the reaction products.

the effect of changes to q1 would be the same as changes toZ1. Therefore, the effect of reasonable changes toq1 on the conclusion can be gauged by the effects of other param- eters.

The effect of changes to the Arrhenius reaction-rate parameters was investigated by repeating the alternating HMX and binder simulation, illustrated in figure 5.13, using values of lnZ1 andE1 corresponding to the Arrhenius time to explosion curve with min- imum and maximum gradient (see section3.4). For PBX9501, these changes had very little effect on temperature histories. For EDC37, the temperature histories are illustrated in figure5.20. Using the lower lnZ1 and E1 values slightly lowers the reaction rate in the binder regions of EDC37, but the effect is not very great because the three sets of Ar- rhenius parameters produce similar explosion times in the temperature range of relevance here (∼1000 K). With the higher lnZ1 and E1 values, more reaction occurs in the simu- lation and even some of the HMX regions react. However, it is clear that detonation still has not been produced by the 1 mm run distance required by the experimental Pop-plot.

Similarly, a representative-geometry EDC37 simulation with these higher values of lnZ1andE1showed increased reaction in the binder but not in the HMX regions; certainly detonation did not occur by the end of the simulation. In section3.4, it was noted that the uncertainty in the Arrhenius reaction rate parameters for the binders is much greater than for HMX. If the binder reaction kinetics are set to be so fast that it reacts to completion almost immediately, using E1 = 160 kJ/mol and lnZ1 = 20.0 (for Z in µs−1), there is still very little reaction in the HMX regions. It is the shock temperature reached in the HMX, along with the HMX reaction rate parameters, that determines the rate of reaction in the HMX. These simulations demonstrate that uncertainties in the Arrhenius reaction parameters do not change the conclusion since no detonation was observed to occur within the 1 mm required by the extrapolation of the experimental Pop-plot.

Overall, the calculations in this section have demonstrated that the uncertainties in the material properties do not affect the conclusion of this chapter, that shock heating of crystals and binder is not a feasible hotspot mechanism in the HMX-based explosives PBX9501 and EDC37.

Figure 5.20: Temperature histories for alternating HMX and binder calculations of EDC37, to investigate the effect of the uncertainty in Arrhenius parameters. Values used wereE1 = 140 kJ/mol and lnZ1 = 11.6 for HMX and lnZ1 = 12.4 for binder (above), andE1 = 160 kJ/mol and lnZ1 = 13.8 for HMX and lnZ1 =14.6 for binder (below), for

5.4

Summary

The Peruse hydrocode, with Arrhenius chemistry and heat conduction, has been used to investigate the feasibility of shock heating of crystals and binder as a hotspot mechanism. Multi-region HMX and binder calculations have been performed at a shock pressure of 20 GPa. At this pressure, the experimental Pop-plot can be extrapolated to give an ex- pected run-to-detonation distance of 1 mm and run time of∼0.15µs. None of the simula- tions has produced detonation within this length or timescale, and there is no reaction at all in lower pressure simulations, demonstrating that shock heating of crystals and binder is not a feasible hotspot mechanism. It is believed that this is the first time this has been explicitly demonstrated. It has been checked that this conclusion is not dependent on the geometry or meshing used for the simulations, and that reasonable changes to the material properties data do not change the result significantly.

Critical hotspots and flame propagation

In this chapter, the models constructed for HMX and the binders in PBX9501 and EDC37 will be used to determine critical hotspot criteria and investigate mechanisms for reaction to spread outwards from hotspots. Section6.1 will describe critical hotspot simulations to examine the behaviour of hotspots in Peruse, to gauge the effect of changes to the ini- tial temperature, geometry and mesh density on the results, and to investigate whether a thin layer of binder can impede or enhance the spread of reaction through HMX. Criti- cal hotspot criteria for HMX and the binders will be compared to results in the literature in section6.2. Flame propagation data will be used to validate the reaction propagation speed in hotspot simulations. Section 6.3 will describe diamond anvil cell experiments and initial simulations to establish the sensitivity to uncertainties in material properties data. The results from a series of flame propagation simulations in Peruse will be com- pared to the experimental data in section 6.4 and suggestions will be made for improv- ing the simulations in future. The implications of the observed reactive wave speed for hotspots in PBX9501 and EDC37 will be discussed in section6.5.

6.1

Critical hotspot calculations

Previous studies of critical hotspots in explosives [24, 25, 26] have modelled hotspots of uniform high temperature, surrounded by cooler regions of uniform lower temperature. This is a simple configuration that allows comparison between different studies and is broadly representative of a variety of hotspot mechanisms, so it will be used through-

Reflective Hotspot diameter r ₂ Effective geometry boundary Hotspot separation

extends to either side Simulated geometry

Reflective boundary

T₁ T₂

1 r

Figure 6.1: Plane geometry used to investigate critical hotspots. The reflective boundary conditions mean that the simulated geometry effectively extends to either side of the com- putational domain. Physically, an infinite series of planar hotspots are being modelled, with hotspot diameter 2r1 and separation 2r2.

out this chapter. The computational geometry comprises a hot region of temperatureT1 and thickness r1, surrounded by a cooler region of temperature T2 and thickness r2. In spherical geometry, this corresponds to a hot sphere surrounded by a cool shell. In plane geometry, it represents a line through the microstructure of the explosive perpendicular to a planar hotspot, as sketched in figure6.1. Reflective boundary conditions are used at each end of the computational domain. It is widely assumed that hotspots in plastic-bonded explosives occupy only a small proportion of the microstructure, so thatr1 < r2. For the initial calculations in this section, geometries withr1 ∼ r2are used to minimise the com- putational expense. Peruse is used to solve the Lagrangian reactive-flow equations 2.7 neglecting species diffusion, with material models from chapter 3. Plane geometry is used for the majority of this chapter; the effect of switching between plane, cylindrical and spherical geometry is investigated in section6.1.6.

The results from an initial calculation with r1 = r2 = 1µm are shown in figure 6.2. Virtual gauges were positioned at 0.1µm intervals through the HMX and their temper- ature histories are shown by the different curves in the top graph. The mesh size was 0.05µm; it will be demonstrated in section6.1.7that this is mesh-converged. Initial con- ditions representative of a post-shock state were imposed on the geometry by specifying the values of specific internal energy and density at the start of the hydrocode calcula- tion. The values in the cooler region were chosen to correspond to a point on the HMX Hugoniot withT2 = 1022 K. The hotspot was given the same initial density but elevated specific internal energy, corresponding to a temperature ofT1 = 1500 K. These states are consistent with constant-volume heating within the hotspot, an assumption that will be

Figure 6.2: Planar HMX hotspot calculation with r1 = r2 = 1µm, T1 = 1500 K and

T2 = 1022 K, as an example of the typical behaviour of hotspot simulations in Peruse. Temperature histories from gauges positioned at 0.1µm intervals through the geometry are shown above, with a pressure trace below.

tested in section6.1.5.

As shown by figure 6.2, the first computational zones to react are those within the hotspot. These start at temperature T1 = 1500 K and react to completion in ∼0.05µs, as determined by the Arrhenius reaction delay time [179]. As these zones react, their temperature rises dramatically causing a reactive wave to propagate outwards into the cooler HMX surrounding the hotspot until, by ∼0.6µs, all the HMX has reacted. The wave propagation speed can be calculated by examining the time at which each gauge rises above a given temperature. The wave speed of∼1 m/s will be discussed in more detail below. There is a small temperature rise in the cooler HMX when the hotspot reacts, caused by a general pressurisation of the system to 35 GPa when the hotspot converts from solid unreacted explosive to gaseous reaction products. The pressure plot shows that the system continues to pressurise as the reactive wave spreads from the hotspot into the cooler HMX. Only one pressure history is shown in figure 6.2 because the traces from all the different gauges overlay each other. This is because the computational domain is so small (only 2µm in extent) that the wave reverberation time is very short (< 1 ns) and approximate pressure equilibrium is maintained at all times.

The reactive wave is driven by heat conduction, as will be demonstrated by comparing the calculation with normal thermal conductivity (figure6.2) to one with reduced thermal conductivity. Figure6.3shows the results of a simulation where the thermal conductivity

kof HMX was reduced by a factor of 10. In both calculations, the hotspot reacts promptly and there is a small temperature rise in the cooler HMX as the system pressurises. How- ever, the propagation speed of the reactive wave into the cooler HMX is significantly lower in the calculation with reducedk, indicating that the reactive wave is driven by heat conduction. It has been suggested that a hydrodynamic mechanism might contribute to the reactive wave [53]; shocks generated by exploding hotspots would heat the cooler surrounding explosive, enhancing its reaction rate. However, this effect is not observed in figure6.2because the wave-reverberation time is much quicker than the hotspot reaction time as will be discussed below.

To investigate whether different behaviour would be produced if the hotspot reacted more quickly, or if the wave reverberation time were longer, two more calculations were run. Figure6.4shows results for the first 2 ns of the simulations. In both calculations, the hotspot temperatureT1 = 5000 K is considerably higher than is likely to be achieved in a real explosive microstructure. It was chosen to yield rapid reaction; the burn fraction plot in figure6.4shows that 50 % reaction is achieved within∼0.1 ns in the hotspot. The

Figure 6.3: Planar HMX hotspot calculation with r1 = r2 = 1µm, T1 = 1500 K and

T2 = 1022 K as in figure6.2, but with the thermal conductivitykof HMX reduced by a factor of 10, to demonstrate that the reactive wave is driven by heat conduction.

pressure and temperature traces in figure6.4 show that the rapid reaction in the hotspot drives a pressure wave through the cooler surrounding HMX. For the blue traces, the computational domain is so small that the wave is reflected from the boundary before reaction has completed. The black traces correspond to a longer computational domain, and demonstrate that reaction in the hotspot causes a ∼60 K jump in the temperature of the surrounding cooler HMX. However, this temperature jump is insufficient to cause any significant reaction overµs timescales and no reaction is observed in the burn fraction plot outside the hotspot. This indicates that a hydrodynamic mechanism of reaction spreading does not occur in these simulations.

The following observations can be made from the initial simulations:

• A wave driven by heat conduction is the only mechanism by which reaction spreads from a hotspot into the surrounding cooler material.

• If the hotspot is so hot that it reacts almost instantaneously, a shock can be driven into the surroundings, but it is only responsible for a ∼60 K rise in temperature (which doubles to ∼120 K when the wave is reflected from the boundary). This is insufficient to cause significant chemical reaction in the cool HMX over µs timescales.

Figure 6.4: HMX hotspot calculations withT1 = 5000 K, T2 = 1022 K and r1 = 1µm. The cooler explosive has thicknessr2 = 1µm for the blue traces and r2 = 9µm for the black traces. This configuration was designed to investigate whether hotspot simulations would behave differently if the hotspot reacted more quickly, or if the wave reverberation time were longer, than in typical simulations.

• If the hotspot has a more realistic initial temperature, it reacts relatively slowly. Although a compression wave propagates into the cooler surroundings, this leads to only a small rise in temperature.

• In most cases, the wave reverberation time is so small that approximate pressure equilibrium is maintained throughout the computational domain.

The following sections will investigate the effects of hotspot temperature, background temperature, hotspot size, background size, initial conditions, geometry and mesh density on the results of hotspot simulations in Peruse. Material properties for HMX and the binders in PBX9501 and EDC37 are taken directly from chapter3. Uncertainties on the data are not considered, since the aim of this section is to establish how the baseline models perform.