A high-density, high-temperature mixture model Page: 3 of 29
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We will rirst describe the physical basis and theoretical tools
needed for building the mixture model. We then apply the mixture model
to binary and ternary mixtures, and to more complex mixtures which are
important in understanding the detonation behavior of condensed
explosives and the possible states of a planetary interior.
Quantum Mechanical Potentials
A reliable mixture model must give the equation-of-state (EOS)
properties of each chemical species in the mixture. This in turn
requires information about the intermolecular potential of each chemical
species. The most rigorous way of getting this information is to do
quantum mechanical calculations. Ideally, we could carry out such
calculations for any cluster of molecules, but in practice such
calculations become unwieldy for all but the simple molecules.
The hydrogen molecule, with only two electrons, comes very close to
the ideal simple molecule. The nitrogen with 14 electrons, has much more
complicated electronic interactions. However, in a high-pressure or
high-temperature environment, only the strongest forces must be specified
accurately, and we can tolerate approximations to the weaker forces. We
can obtain a reasonably reliable calculations of the quantum mechanical
potential for nitrogen molecules by using the self-consistent field (SCF)
Figure 1 compares SCF intermolecular potentials of two H29 and two
N210 molecules at four different molecular orientations. For hydrogen,
each of the orientations gives the same curve (indicating a highly
isotropic nature of the interaction), but for nitrogen, the curves are
far apart. At an intermolecular spacing of 2.9 A (about that of the
highest shock compression achieved in Ref. 12a from an initially
unshocked liquid state), the difference in energy between the X and L
geometries (defined in Fig. 1) is about 200 times larger for nitrogen
than for hydrogen.
Spherical Potentials for Like-Pair Interactions
The ab initio quantum mechanical potentials described above are
cumbersome, and we must use simpler potentials (with the same physical
features) for mixture calculations. Fortunately, at high temperatures
molecules can rotate more easily and such orientational ordering as
hydrogen bonding in water can be at least partly broken up. In many
dynamic experiments, the compression is still relatively low and the
temperature is high enough so that the repulsive force appears to be
independent of orientation, even for a highly nonspherical molecule such
as N2. Hence, we need not complicate the expression by explicitly
allowing for nonsphericity. A theoretical justification of the spherical
approximation has been provided by Shaw, Johnson, and Holian6a and
Lebowitz and Percus.6b They showed that the fluid EQS of a system
interacting with a nonspherical potential V(r,-1,w2) can be approxi-
mated by those of a fluid with a spherical potential V(r) equal to the
angular median of V(r:,,0l2)-
For a high-pressure and high-temperature application, a spherical
potential must be able to describe three essential characteristics: the
range of interaction, the depth-of-attraction, and the stiffness of the
repulsion. The simplest physically realistic potential which satisfies
the above requirement is an exponential-6 (exp-6) potential,
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Ree, F.H. A high-density, high-temperature mixture model, article, March 1, 1988; California. (https://digital.library.unt.edu/ark:/67531/metadc1059100/m1/3/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.