Computational Analysis of Nitramine Derivatives &
Carbon Analogs as Potential HEDMs
Abstract
The purpose of this project is to determine the thermochemical
properties of the molecules of interest in order to evaluate them as
explosive materials and determine if they meet the requirements to be
classified as high energy-density materials (HEDMs) while comparing the
carbon and nitrogen skeletons.
The six molecules of interest are trinitramide (N1), tetranitrotetrazetidine
(N2), hexanitrohexazinane (N3), trinitromethane (C1),
1,2,3,4-tetranitrocyclobutane (C2), & 1,2,3,4,5,6-hexanitrocyclohexane (C3).
The Gaussian program is selected to run the computations for this
project due to its popularity and use by experts in this field.2-4,16
Calculations are performed with the Density Functional Theory at the
B3LYP level using the 6-311G** basis set as previous studies have found
satisfactory results with this theory and basis set .2-4,8,9,16
Formation enthalpies and other thermal properties were obtained from
the computational output. The
oxygen balance was used to determine the detonation products, and the heats
of detonation were used to evaluate the detonation performances of the
molecules.
1. Introduction.
Computational methods are procedures that use simple algorithms to evaluate
complex problems and analyze large data sets.
They employ computers to reduce the expense of lengthy calculations,
increase the reliability of analyzing various sizes of data sets, and
provide easier accessibility by graphing large data sets.
The use of computational methods began when advancements in
mathematics, and its various application fields produced more complex
formulas to accurately describe the various phenomena.
Today, computational methods are used in a variety of fields
including both the natural and social sciences as well as mathematics.
A few of the numerous applications are drug design, weather
forecasting, evaluating proposed syntheses, protein or DNA/RNA sequencing
and sequence comparisons, molecular modeling, and calculating atomic and
molecular properties using quantum mechanics.
The accuracy of the computed values based on quantum mechanics
compared to the experimental values depends on the level of theory and basis
set used and varies from program to program for a property of interest.
In general, the freely available programs are accurate enough to
provide qualitative data at relatively low levels of theory, but higher
levels of theory are required to obtain quantitative data and not all of the
programs have this capability.
In general, the commercially available programs seem to be more
comprehensive and capable than the freeware programs.
Quantum mechanical computational methods are used in this project to
investigate the explosive properties of three nitramine based molecules and
their carbon skeleton analogs.
The principal interests of this research were to determine if any of the
molecules fit the criteria of high-energy density materials, have potential
as a propellant or an explosive, and whether a carbon or nitrogen skeleton
yields better performance.
An explosion is a rapid and violent
release of mechanical, chemical, or nuclear energy accompanied by a
physical, and usually audible, shockwave, the generation of high
temperatures, the release of gases, and sometimes light resulting from a
rapid physical, chemical, or nuclear change.12,13
There are three types of explosions: physical, chemical, and atomic.
Of these three types, this research project involves chemical
explosions. A chemical
explosion is
a chemical reaction or change of state which occurs over an exceedingly
short frame of time with the generation of a large amount of heat and
generally a large quantity of gas.
Based on their performances and properties, chemical explosives are
separated into three groups: primary explosives, secondary explosives, and
propellants.1,10
Materials within the primary and secondary categories can also be classified
as high energy-density materials (HEDMs) if they, ideally, have
a density of 1.9 g cm-3, a detonation velocity of 9.0 km s-1,
and a detonation pressure of 40.0 GPa (approximately).2
Primary explosives are quite sensitive to shock, friction, electric sparks,
and high temperatures; consequently, they will detonate when subjected to
any stress. Upon detonation,
the material’s molecules dissociate and produce a tremendous amount of heat
and shock which, in turn, initiates a second more stable explosive.
In other words, they rapidly transition to detonation from burning
and are capable of transmitting the detonation wave to less sensitive
explosives (i.e. secondary explosives).
As a result of their capacity to initiate a secondary explosive in
addition to their sensitive nature, their ability to explode regardless of
confinement, and a general detonation velocity range of 3,500 to 5,500 m s-1,
primary explosives are frequently used in ‘initiating devices’.1,10
Materials that fall into the
secondary explosives (a.k.a. high explosives) category cannot be readily
detonated by heat or shock and are generally more powerful than primary
explosives. The secondary
explosives can only be detonated by the shock produced from the explosion of
a primary explosive. Upon
initiation, they almost instantaneously dissociate into more stable
components and detonate with a general velocity range of 5,500 to 9,000 m s-1.1,10
The third class of chemical explosives are propellants which are
combustible materials containing enough oxygen to achieve complete
combustion. While they can be
initiated by a spark or flame just like primary explosives, the key
differences between propellants and the other explosives are that they burn
but do not explode and change from a solid to a gaseous state relatively
slowly (a few milliseconds compared to a millisecond or less).1,10
When an explosive reaction takes place, the explosive material
atomizes in an extremely exothermic process and at an extremely high rate.
The atomization products immediately assemble into small stable
molecules; generally forming carbon monoxide, carbon dioxide, water, and
nitrogen gas. The products of
the decomposition reaction are determined by the material’s oxygen content
which is described by its oxygen balance (%OB).
This quantity describes the amount of oxygen, in weight percent,
liberated from the material by complete oxidation of the atomization
products.1,5,10
If the %OB is greater than or equal
to zero, the explosive undergoes complete combustion and the heat released
during the formation of the gas products is the heat of explosion; however,
if the %OB is less than zero, incomplete combustion takes place giving rise
to products like carbon monoxide, solid carbon, and hydrogen gas with a less
amount of energy released as the heat of explosion.
The material’s detonation reaction follows one of the two following
sets of rules: the Kistiakowsky-Wilson Rules (K-W rules) and the Modified
Kistiakowsky-Wilson Rules.1,10,25
|
Table 1: Detonation Reaction Rules1,10,25 |
|||
|
Kistiakowsky-Wilson Rules |
Modified Kistiakowsky-Wilson Rules |
||
|
Rule |
Condition |
Rule |
Condition |
|
1 |
Carbon atoms are converted to carbon monoxide |
1 |
Hydrogen atoms are converted to water |
|
2 |
If
any oxygen remains then hydrogen is then oxidized to water |
2 |
If any oxygen remains then carbon is converted to carbon monoxide |
|
3 |
If
any oxygen still remains then carbon monoxide is oxidized to carbon
dioxide |
3 |
If any oxygen still remains then carbon monoxide is oxidized to
carbon dioxide |
|
4 |
All the nitrogen is converted to nitrogen gas, N2 |
4 |
All the nitrogen is converted to nitrogen gas, N2 |
The set of rules used is determined by the material’s %OB.
The K-W rules are used with materials ranging from oxygen abundant to
moderately oxygen deficient (an %OB greater than – 40.0), while the Modified
K-W rules are used for severely oxygen deficient materials (an %OB less than
– 40.0).1-3,5,16,25
The
properties essential for the evaluation of the theoretical compounds are the
detonation velocity, detonation pressure, and relative stability.
Because the reaction occurs in approximately 0.01 s and gases cannot
instantly expand, there is a fraction of a second where the gas remains
inside of the container with constant volume.
The extreme heat produced by the explosion and the small volume of
the gaseous products yield a pressure great enough to create a shock/blast
wave.1,10
The rate at which this wave propagates is called the detonation
velocity.6 The
detonation pressure is the pressure produced by the reaction in the fraction
of a second before the volume changes.20-23
The relative stability is the compound’s stability based on the
calculated Gibbs free energy of formation of the compound relative to that
of a reference compound which is known to be stable.
The
determination of these properties depends on a number of parameters
including theoretical density, Gibbs free energy of formation, zero-point
energy, enthalpies of formation and detonation, molecular volume, and oxygen
balance.2-5,8,9,15
In this project, the theoretical parameters and properties essential
for evaluation of explosive power are determined using the well-established
equations and computed molecular properties for the compounds shown in
Figure 1.
This project was selected after inquiring about the structure of
1,3,5-trinitroperhydro-1,3,5-triazine
(a.k.a. RDX), the explosive compound in the military grade plastic explosive
C-4.
Because of the computational expense and the limited computing power
available, a six-membered ring was determined to be the largest molecule to
be studied. Due to the amount
of time spent attempting to initially build RDX, it was decided that the
study would be limited to three molecule sizes.
With RDX being composed of methylene and nitramine groups, a fully
nitrated nitramine was selected as the smallest aza molecule.
Four-membered rings were chosen to be the third group of molecules
intermediate in size between the nitramine and the six-membered ring
molecules.
|
Group 1 |
Group 2 |
Group 3 |
|
|
|
|
|
Trinitramide
N1 |
Tetranitrotetrazetidine
N2 |
Hexanitrohexazinane
N3 |
|
|
|
|
|
Trinitromethane
C1 |
1,2,3,4-Tetranitrocyclobutane
C2 |
1,2,3,4,5,6-Hexanitrocyclohexane
C3 |
|
|
|
|
|
Nitromethane
R1 |
1,3,3-Trinitroazetidine
R2 |
1,3,5-Trinitroperhydro-1,3,5-triazine
R3 |
Figure 1: Structures of Cyclic Nitramide Derivatives (N),
Carbon-Skeleton Analogues (C), and Similar Skeletal References (R).
2. Computational Methods
For this project, a number of calculations were executed including Geometry
Optimization2-4, Bond Order2,4,26, Monte Carlo
integration2-4,16, Frequency (thermochemical analysis) and
Vibrational Analysis2,3,26.
A variety of computational methods have been used for these
calculations by other researchers, but the density functional theory (DFT)
at the B3LYP level2-4,8,9 (DFT-B3LYP), and the quantum chemical
composite methods G2 (Gaussian 2) & CBS-Q (Complete Basis Set Q)2
seem to be the most common.
Other researchers have successfully used the basis sets 6-31G(d)3,16
and 6-311G(d,p)2-4,8,9 for geometry and thermodynamic data.
Computational engines such as Gaussian2-4, LOTUSES (Linear
Output Thermodynamic User-friendly Software for Energetic Systems)5,
ChemBasis3D, and MOPAC are commonly used for this type of research.
2.1 Geometry Optimization
All geometry optimizations were carried out at the DFT–B3LYP level
using the STO-3G, 3-21G, & 6-31G(d) basis sets sequentially.
The last job executed for each final structure was a DFT–B3LYP
optimization at 6-311G(d,p). To
simplify and shorten the process of building the molecules, an appropriate
base structure was modified stepwise to construct the target molecule.
For example, when converting hydrogen atoms of the base structure
into amino groups, the hydrogen atom was changed to a nitrogen atom and the
‘Add Hydrogens’ tool was used to add the hydrogen atoms and form the amino
group. Conversion of amino
groups into nitro groups was done by changing the hydrogens to oxygens and
creating the double bonds. When
carbon atoms were converted to nitrogen atoms, the equatorial hydrogen was
deleted and when a carbon‘s hydrogen was changed into a nitrogen, the axial
hydrogen was changed to nitrogen.
2.1.a Group 1 Molecules.
Starting with the central atom, the base molecule was created using
the ‘Add Hydrogens’ button. The
appropriate amount of hydrogen atoms was then changed into nitro groups.
2.1.b Group 2 Molecules.
Cyclobutane was imported from WebMO's structure library.
The appropriate carbon atoms were changed into nitrogens and the
structures were optimized. The
appropriate hydrogens were then changed into nitrogen atoms and the average
central atom to NO2 bond parameters of the respective Group 1
molecule were used to make the nitro groups.
Before the structures’ final optimization, the dihedral angles formed
by the two ring atoms and the N=O for each nitro group were balanced by
setting them equal to the average of their absolute magnitudes with the
appropriate sign.
2.1.c Group 3 Molecules.
Cyclohexane was imported from WebMO’s structure library.
After each alteration, the structures were optimized.
First, the appropriate carbon atoms were changed into nitrogens, then
the hydrogens on three non-adjacent atoms in the rings were changed into
amino groups. Next, the
structural parameters of the amino groups were averaged for each molecule
and used to create hexamines.
In the last two steps, the hexamines were converted into
1,3,5-trinitro-2,4,6-triamines using the structural parameters of the nitro
groups in the respective Group 2 molecules followed by conversion to
hexanitrated rings. RDX
(1,3,5-trinitroperhydro-1,3,5-triazine) was built via the repetitive
conversion of a ring nitrogen's hydrogen atom into an amino group and use of
the structural parameters of the N-NO2 group in TNAZ
(1,3,3-trinitroazetidine).
2.2 Bond Order
All bond orders were calculated at the DFT B3LYP level with the
6-31G(d) basis set. This
calculation predicted which bond in the molecule is the weakest and would be
severed first. This information
was then used to eventually evaluate the relative thermal stability of the
molecules via their bond dissociation energy (BDE).
The BDE is defined as the reaction enthalpy of the bond scission
reaction at 1 atm and 298 K.
The general formula for the bond scission reaction of the molecule
A–B
is as follows:
|
|
A–B(g) → A∙(g) + B∙(g) |
1 |
The enthalpy of this reaction is found using equation 2.
|
|
|
2 |
where
is
the formation enthalpy of the radical
A∙, 
is
the formation enthalpy of the radical
B∙,
and
is
the formation enthalpy of the molecule.
These enthalpy values were calculated by the Thermochemistry
calculation in the MOPAC engine of WebMO using the PM3 (Parametric Basis Set
3) basis set which has been found to produce acceptable thermodynamic
data.2,3,4,26
2.3 Monte Carlo Integration
The Volume job type uses a Monte Carlo integration to calculate the
molecular volume of the isolated gas phase molecule, which is defined as the
space enclosed within the 0.001 electrons/bohr3 electronic
isodensity surface. The volume
calculated by Gaussian is only accurate to two significant figures, but by
using the option Tight with the
Volume calculation, an increased density of points is used for the
integration which increases the accuracy to ~10%.
As a result, the generally used method takes the average of 100
single point calculations.2,3,4,16,20,27
2.4 Frequency and Vibrational
Analysis
The Vibrational Analysis was used to determine if the optimized
structure was at a local minimum or at a global minimum on the molecule’s
potential energy surface.
Negative frequencies found by the vibrational analysis indicated that the
structure was at a local minimum; If these were found, adjustments would
have to be made to the structure followed by another optimization and
vibrational analysis.
The Frequency job was then used to calculate the thermochemical
properties of the molecules.
The quantities used to evaluate HEDMs were then calculated from the output
data of this job.
After geometry optimization using DFT-B3LYP/6-311G(d,p) and a
vibrational analysis to ensure that the optimized structures are at the
global energy minimum on the potential energy surface2-4, the
HEDM parameters are calculated.
The results are evaluated to classify the molecules of interest into one of
the three categories of chemical explosives and their properties were
considered to determine whether or not they qualify as HEDMs.
3. HEDM Parameters
The
properties used to evaluate the theoretical materials are the theoretical
density, detonation velocity, and detonation pressure as well as the
relative thermal stability.
Figure 2 shows that the heat of formation is the most important quantity
used in the evaluation, but the evaluation depends on a number of other
parameters including total molecular energy, zero-point energy, enthalpy and
detonation, molecular volume, and oxygen balance.2-5,8,9,15
Figure 2: HEDM Parameter Relations18
3.1 Heat of Formation
3.1.a Methods
The heats of formation of reference compounds (RDX, TNAZ, and nitromethane)
were calculated and used to find the heats of detonation.
Several different methods can be used to determine a theoretical
molecule’s formation enthalpy.
These methods use different types of reactions that aid in the cancellation
of systematic errors that arise in quantum mechanical computations.
Atomization reactions compare the absolute energy of the molecule to
the absolute energy of its constituent atoms with known ∆Hf.
Isodesmic reactions conserve the number of types of bonds.
Homodesmic reactions conserve both the number of types of bonds and
the hybridization of the atoms.17
Isogyric reactions conserve the number of electron pairs.19
It is acknowledged that the use of homodesmic reactions would yield
more accurate results. However,
since the procedure for their use from the output data was unclear, the
atomization method, described by the White Papers and Technical Notes
provided by Gaussian Inc.29, was used.
3.1.b Atomization Method Calculations
The first step was to calculate the atomization energy of the
molecule,
,
by the following equation for a molecule composed of x many atoms of X:
|
|
|
3 |
where
is
the molecule’s total electronic energy,
is
the molecule’s zero-point energy, and
is
the total electronic energy of the atoms.
In the next step, the formation enthalpy of the molecule at 0 K,
,
was calculated by combining the formation enthalpy of the atoms at zero
Kelvin,
,
with the molecule’s atomization energy.
|
|
|
4 |
Finally, the formation enthalpy of the molecule at 298 K,
,
was calculated using equation 5:
|
|
5 |
where
is
the correction to the thermal enthalpy of the molecule from 0 to 298 K, and
is
the thermal correction to the enthalpy of the atoms from 0 to 298 K.
3.2 Oxygen Balance
A material’s oxygen balance describes the amount of oxygen, in weight
percent, liberated from the material by complete oxidation of the
atomization products1,5,10,25 and is an important criteria used
in classifying potential HEDMs15.
The %OB of an explosive of the generic form CwHxNyOz
is calculated using equation 6:
|
|
|
6 |
where
is
the molar mass of oxygen, which is set to 16 g/mol, and
is
the molar mass of the explosive.1,10
The %OB value was then used to determine the ideal detonation
reaction each material would undergo.1-3,5,16,
3.3 Detonation Performances
The detonation velocity (D) and detonation pressure (P) were
calculated using the Kamlet-Jacobs equations,2,3,4,14,20,26
|
|
|
7 |
|
|
|
|
|
|
|
8 |
where N is the moles of ideal gaseous detonation products per gram of
explosive, M is the average molecular mass of the gaseous products (g/mol),
Q is the enthalpy change of the detonation reaction (cal/g), and ρ is the
density of explosive (g/cm3). The
theoretical density was obtained from the molecular weight and the
theoretical molecular volume which is generally defined as the space within
an electronic isodensity contour of 0.001 electron/bohr evaluated using a
Monte Carlo integrator.2-4,8,9,14,16
The generally accepted method for obtaining the molecular volume is
to use the average molecular volume of 100 single point calculations2,16;
however, due to time constraints and resource availability, the volume
produced by a single calculation was used.
4. Results & Discussion
4.1 Heats of Formation
The magnitude of the heat of formation is a good indicator of the
amount of energy a HEDM contains relative to its free elements and its
accurate prediction is very important in calculating the detonation
pressures and velocities. Table
1 contains the relevant thermodynamic data computed by Gaussian and the
calculated thermochemical properties for the six molecules of interest and
the three reference compounds. Table
2 contains the estimated formation enthalpies of the molecules.
In each group, the ΔHf increases with the number of NO2
groups. At 298 K, the nitrogen
skeleton molecules have higher formation enthalpies than the carbon skeleton
molecules and the reference molecules.
Thus, N–NO2 bonds contain more energy than C–NO2
bonds and yield significantly higher ΔHf values.
|
Table 1. Computed and Calculated Thermodynamic Values |
||||||
|
|
Computed Values |
Calculated Values |
||||
|
Hartree/Part. |
kcal/mol |
|||||
|
ZPEcorr |
Hcorr |
∑(Ԑ0+Hcorr) |
Ԑ0 |
Atomization |
∆Hf(M,
298 K) |
|
|
N1 |
0.040261 |
0.050313 |
-670.0919 |
-670.1422 |
732.1888 |
93.0453 |
|
N2 |
0.068045 |
0.083213 |
-1039.3641 |
-1039.4473 |
1170.7376 |
237.0028 |
|
N3 |
0.103915 |
0.126683 |
-1559.0867 |
-1559.2134 |
1781.5534 |
331.2266 |
|
C1 |
0.054277 |
0.064133 |
-654.0998 |
-654.1639 |
889.6246 |
53.1414 |
|
C2 |
0.119986 |
0.134332 |
-975.3522 |
-975.4865 |
1773.2088 |
102.0482 |
|
C3 |
0.183990 |
0.206085 |
-1463.0534 |
-1463.2595 |
2675.9565 |
139.8074 |
|
R1 |
0.049637 |
0.053833 |
-245.0367 |
-245.0917 |
567.1550 |
16.2960 |
|
R2 |
0.106048 |
0.118245 |
-786.8300 |
-786.9483 |
1465.3301 |
114.2016 |
|
R3 |
0.141725 |
0.155324 |
-897.5234 |
-897.6787 |
1773.8602 |
153.1587 |
|
Zero-Point Corrections (ZPEcorr), Thermal Corrections
to Enthalpy (Hcorr),
Total Electronic Energies (Ԑ0),
Sum of Total Electronic Energies and Thermal Corrections to
Enthalpy (∑(Ԑ0+Hcorr)),
Atomization Energies, & Formation Enthalpy(∆Hf) |
||||||
|
Table 2: Estimated Formation Enthalpies in the Gas Phase, in
kcal/mol |
||
|
|
∆Hf
(0 K) |
∆Hf
(298 K) |
|
N1 |
58.9 |
68.6 |
|
N2 |
180.3 |
197.1 |
|
N3 |
244.9 |
270.3 |
|
C1 |
-10.9 |
-1.7 |
|
C2 |
31.1 |
45.4 |
|
C3 |
40.3 |
62.6 |
|
R1 |
-19.02 |
-14.5 |
|
R2 |
15.4 |
27.4 |
|
R3 |
45.3 |
59.3 |
|
Corrections were made using scaling factors based on the
estimated difference between
∆Hf, 0K
and
∆Hf, 298K
as determined from computed Cv values. |
||
4.2 Detonation Properties
The detonation velocity, detonation pressure, and density are the
three properties essential for evaluating energetic materials.
If the values of these parameters meets or exceeds the theoretical
ideal values for an HEDM, then the material is classified as such.
The detonation velocities and pressures, along with the parameters
involved in their calculation are shown in Table 3.
The average molar mass of the
gaseous detonation products (M), the moles of gaseous detonation products
produced per gram of explosive (N), and the enthalpy of detonation (Q) were
determined from the detonation reactions based on the material’s oxygen
balances in Table 4.
