3.1. Isotherms and Isobars
Although the main topic of the present study is calculation of the elastic coefficients, we also obtain predictions of
-HMX unit-cell volume as functions of pressure and temperature.
Figure 1 shows unit-cell volume
V as a function of pressure
P at 300 K obtained in this work, along with MD results due to Sewell et al. [
14] and experimental data of Yoo and Cynn [
29], Gump and Peiris [
30], and Olinger et al. [
31]. The inset in
Figure 1 shows the lower pressure part of the same data. The red curve in
Figure 1 shows the third-order Birch–Murnaghan isotherm [
32] fitted to our isotherm data. Fitting compression data to isotherms is subtle [
10]. Here, we applied simple, unweighted fits of the Birch–Murnaghan fitting form to the experimental and simulated
data. Our results are close to those of Sewell et al. This is not surprising as they used practically the same force field but with all covalent bonds fixed at constant values. The experimental results of Yoo and Cynn [
29] are in overall qualitative agreement with our data but show slightly higher compressibility of the material. In addition, our simulations do not predict the subtle phase transition reported in [
29] at approximately 27 GPa. Although we have limited results below 10 GPa, the changes in volume that we predict in this region are also similar to the volume changes observed experimentally in [
30,
31]. We provide in the
Supplementary Materials the full set of lattice parameters and unit-cell volumes for all pressures and temperatures studied. The bulk moduli extracted from the experimental isotherm data are compared to our MD results in the next subsection.
Figure 2 shows the five isotherms calculated in the present study. The higher temperature isotherms lie above the lower temperature ones. This behavior is physically reasonable: unit-cell volume increases as the temperature increases due to thermal expansion. The relative volume increase with increasing temperature becomes less pronounced for higher pressures. Otherwise, the isotherms are all quite similar.
Figure 3a shows isobars for the five pressures considered. As expected, the unit-cell volume increases as the temperature increases. The volume increase becomes less pronounced for higher pressure isobars. While the increase in unit-cell volume with increasing temperature is not surprising, the changes in the unit-cell geometry exhibit some interesting features.
Figure 3b–d show the lengths of the unit-cell vectors
,
, and
, respectively. Surprisingly, at 5 and 10 GPa, the length of vector
decreases as the temperature increases. Somewhat similarly, at 20 and 30 GPa, vector
shows almost no change in length as the temperature changes. Similar counterintuitive behavior of some unit-cell lattice vectors was observed experimentally but not emphasized by Gump and Peiris [
30].
Figure 3e shows how angle
(the angle between vectors
and
) changes with temperature and pressure. For a given temperature,
decreases with increasing pressure (the crystal becomes “more orthorhombic”). For a given pressure, the angle increases slightly as the temperature increases.
3.2. Elastic Coefficients
The calculated elastic coefficients and corresponding bulk and shear moduli on the 300, 500, 700, 900, and 1100 K isotherms are listed, respectively, in
Table 1,
Table 2,
Table 3,
Table 4 and
Table 5. There are two clear trends in the results. For a given temperature, magnitudes of the elastic coefficients increase substantially with increasing pressure. For a given pressure, magnitudes of the elastic coefficients decrease with increasing temperature. Both trends are physically plausible: As the crystal is compressed at a given temperature, it is expected to become more stiff and so the elastic coefficients become larger. On the other hand, as the crystal temperature increases at constant pressure, the crystal expands and in doing so becomes less stiff.
As an example,
Figure 4 shows the pressure dependence of the elastic coefficients and isotropic moduli at 300 K (using data from
Table 1). As the pressure changes from 1 atm to 30 GPa, the magnitudes of the elastic coefficients increase by a minimum of about four-fold for
to about 60-fold for
. The Voigt–Reuss–Hill bulk and shear moduli increase by about 12- and 6-fold, respectively. Similar behavior is observed at the higher temperatures (not shown).
The typical temperature dependence of the elastic coefficients is shown in
Figure 5, where the results for pressure 5 GPa are presented. This is the lowest pressure among the ones we studied for which
-HMX remains crystalline for all temperatures on the time scale considered. The elastic coefficients decrease with increasing temperature approximately linearly. The rate at which the magnitude of the elastic coefficients decreases with temperature ranges from about 2.5 GPa per 100 K for
to about 0.07 GPa per 100 K for
. The Voigt–Reuss–Hill bulk and shear moduli decrease with temperature at a rate of about 1 GPa per 100 K and 0.5 GPa per 100 K, respectively. Qualitatively similar temperature dependence of the elastic coefficients was observed at higher pressures but the rates at which the coefficients decrease become smaller.
In
Figure 6, we compare our results at 1 atm and 300 K to the published theoretical and experimental data for the
-HMX elastic coefficients variously at ambient conditions [
7,
8,
9,
14] or zero temperature and pressure [
15,
33,
34]. Our results are, in general, consistent with the MD simulations of Sewell et al. [
14] for which practically the same force field was used but with all the covalent bonds frozen. As a result of the bond constraints, most of the elastic coefficients in [
14] are slightly larger in magnitude compared to our values. Similarly, there is an overall consistency with the MD/energy-minimization results of Mathew and Sewell [
15], obtained with practically the same force field used here but at zero temperature. As can be expected, their values are higher than ours because the crystal stiffens with decreasing temperature, as discussed above. The DFT results in [
33,
34] give higher values of elastic coefficients and elastic moduli compared to ours. This is consistent with the fact that those results correspond to zero temperature and pressure. Note that, although DFT calculations have been used to calculate lattice parameters and unit-cell volumes on the cold curve (see, e.g., [
33]), to our knowledge, they have not been used to predict elastic coefficients at elevated pressures. Moreover, incorporating temperature dependence into such calculations using, for example, the quasi-harmonic approximation would be unreliable due to the need to account for (anisotropic) thermal expansion across hundreds of kelvins; and explicit simulations analogous to, and on the scale of, those studied here are practically infeasible. There is a significant disparity among the various experimental results, which has been attributed to variations in measurement techniques and sample purity. Reanalysis [
6,
14] of the experimental data has shown that this can also result from lack of redundancy in the acoustic velocity measurements and sensitivity of the numerical solution to initial conditions of the multivariate minimization used to extract the elastic coefficients. For example, it is known that only five of the thirteen nonzero elastic coefficients reported in [
7] were accurately determined [
14]. Our results agree reasonably well with the experimental data of Sun et al. [
9], with more pronounced differences compared to the other experimental data.
To the best of our knowledge, there are no experimental results for the elastic coefficients of
-HMX at higher pressures even for 300 K. However, the room-temperature bulk modulus as a function of pressure can be obtained from fits of experimental isothermal compression data to an equation-of-state fitting form (here, the third-order Birch–Murnaghan equation of state), which yields pressure
P as a function of volume
V. The volume-dependent bulk modulus
can be calculated as
. The pressure-dependent bulk modulus
can then be obtained by expressing the volume in
as a function of pressure using the equation of state. This approach is less precise than obtaining
K directly from the elastic tensor at a given
T and
P due both to the assumed form of the equation of state and the typically small numbers of data points on the isotherm available for fitting.
Table 6 lists the room-temperature bulk moduli calculated at five pressures using this approach, for three sets of experimental data [
29,
30,
31] and our calculated 300 K isotherm in
Figure 1. The table also includes the pressure-dependent Voigt–Reuss–Hill bulk moduli reported in
Table 1. Note that the results for [
30,
31] at 10, 20, and 30 GPa represent extrapolations of the fitting form to pressures higher than those for which the corresponding isotherms were measured.
In the table, one can see that the MD-based results (top two rows) are self-consistent in that the bulk moduli computed directly from the elastic tensor and via the equation-of-state fit are in good agreement for all pressures considered. Our values obtained by the direct approach are slightly larger than those obtained using the data from Yoo and Cynn [
29] for all pressures except 30 GPa, for which they are about the same. Our estimates based on the data from [
30,
31] are less reliable because, as noted above, the pressure states considered in those references were below 8 GPa. Nevertheless, there is good agreement between our results and those obtained based on the work of Gump and Peiris [
30] with the exception of
P = 1 atm. Our values are lower than those obtained from the data in [
31] for all pressures except
atm, for which our value is larger.