MOLECULAR DYNAMICS SIMULATION OF DISSOCIATION KINETICS AIAA Paper 2000-0213 Presented at the AIAA Aersospace Sciences Meeting Andrew L. Kantor, Lyle N. Long, and Michael M. Micci Department of Aerospace Engineering The Pennsylvania State University University Park, PA 16802 1 MOLECULAR DYNAMICS SIMULATION OF DISSOCIATION KINETICS Andrew L. Kantor, Lyle N. Long, and Michael M. Micci Department of Aerospace Engineering The Pennsylvania State University University Park, PA 16802 Abstract q = generalized coordinates r = position vector The vibrational energy distribution and the degree of re = mean bond length dissociation within a system of hydrogen and oxygen rij = position vector between atoms i and j molecules was modeled using molecular dynamics t = time (MD). The first step in this process was to model the T = temperature atomic and molecular interactions. Since hydrogen and u = intermolecular potential oxygen form diatomic molecules, vibration is the only v = velocity vector intramolecular force that must be computed. The Morse V = intramolecular potential potential is used to perform this calculation. Atomic α = degree of dissociation interactions outside the molecule are modeled using the β = Morse parameter Lennard-Jones potential. The vibrational energy level ε = Lennard-Jones energy parameter distribution of this model demonstrated excellent εv = vibrational energy agreement with the Boltzmann distribution. In this ν = quantum energy level molecular dynamics simulation, dissociation occurs θd = characteristic dissociation temperature when the potential energy between two vibrating atoms ρ = density exceeds a critical value. Recombination is also possible ρd = characteristic density between two previously dissociated atoms by the σ = Lennard-Jones size parameter reverse mechanism. This process enables a system to ω = spacing of vibrational energy levels start in a state of molecules and proceed to an e x ω = first anharmonic correction equilibrated state of atoms and molecules. The e e molecular dynamics simulation accurately modeled both the rate of dissociation and the ratio of species at Introduction equilibrium. Finally, a system of both hydrogen and oxygen was simulated. The results of this simulation The ability to model a combustion process were compared to the CET code and showed excellent numerically is very important to the development of agreement. This investigation demonstrated that simple future propulsion systems. For example, a computer chemical reactions in relatively large systems can be model would be useful in cases where an analytical modeled using molecular dynamics. solution was not valid or an experiment was too difficult to perform. One method of modeling chemical Nomenclature reactions through computer simulation is molecular dynamics. The first step in simulating a reacting gas by a = acceleration vector molecular dynamics is determining how vibrational C = rate constant f energy is distributed among the atoms. Once this is D = dissociation energy accomplished, modeling simple reactions is possible. E4 = Morse parameter One of the least difficult reactions to model is the f = intermolecular force dissociation and recombination of a homonuclear g = degeneracy l diatomic species such as hydrogen and oxygen. H = Hamiltonian k = Boltzmann constant L = Lagrangian Molecular Dynamics m = atomic mass n = rate constant Molecular dynamics is the numerical simulation of N = number of molecules atomic and molecular motions and forces in a system. N’* = number of molecules in a particular energy level By observing the trajectories of these particles, many p = momentum thermodynamic and fluid mechanical properties can be 2 calculated. The main advantage of this technique is that experiments, the intramolecular potential is required to its accuracy is only dependent on the selection of an have a potential well at the mean bond length and intermolecular potential and the precision of the approach the dissociation energy as the atoms become numerical integrator. No physical properties need to be infinitely separated. Also, like the Lennard-Jones known nor assumptions made about the simulated potential, it must go to infinity as the atoms approach medium. However, the disadvantage is that the size of each other. One such model that conforms to these the molecular systems must be extremely small. This is specifications is the Morse potential.2 This potential is due to the large number of force calculations that must given by be made. − β ()− −β ()− Modeling the atomic potentials is one of the most V ()r = De 2 r re − 2De r re + E (3) important parts of a molecular dynamics simulation. ∞ This particular investigation deals with atoms that are free to vibrate within the molecules. Such a system where D is the dissociation energy, r is the internuclear β requires both intermolecular and intramolecular separation, re is the mean bond length and and E∞ are potentials. The intramolecular potential is utilized in constants computed from spectroscopic data. Unlike computing the forces between two atoms of the same Lennard-Jones, the Morse potential is based on four molecule, while the intermolecular potential is used to parameters. The differences between the two potentials calculate forces between atoms of different molecules. can be seen in Figure 1. This figure shows that the depth of the Morse potential well compared to its long Intermolecular Potential range potential is far greater than that of Lennard-Jones. In addition, the Morse potential permits smaller separation distances between the atoms. In molecular dynamics, the potential is taken to be pairwise additive. This means that the total energy in −18 x 10 the system is a sum of the isolated two-body 2.5 contributions. Three body and higher terms are usually neglected because of the vast increase in computational time and their relatively small influence on gaseous 2 systems. The most commonly used pairwise additive potential, u, for modeling intermolecular forces 1.5 Lennard Jones between atoms separated by a distance rij is the one proposed by Lennard-Jones.1 1 Morse Energy (J) 12 6 σ σ u()r = 4ε − (1) ij r r 0.5 ij ij This potential produces a short range atomic 0 1 1.5 2 2.5 3 3.5 repulsion which simulates the overlap of electron −10 rij (m) x 10 clouds and a long range attraction due to induced Figure 1. Morse and Lennard-Jones Potentials dipoles. The only adjustable parameters are a size σ ε parameter, , and an energy parameter, . Both of these Finite Difference Method values are a function of the species being simulated. Since the intermolecular forces are necessarily Once the potentials have been established, the conservative, the force resulting from the above equations of motion can be produced. One typical form potential is of the equations of motion in molecular dynamics is the Hamiltonian.3 The Hamiltonian is defined in terms of 13 7 du()r ε σ σ the Lagrangian L and a set of generalized coordinates q f ()r = − = 24 2 − (2) σ dr r r & & & & ()= − () H p, q ∑ qk pk L q, q (4) Intramolecular Potential k where p is the momentum and is given by The potential between two atoms of the same molecule differs slightly from the intermolecular potential. In order to conform to spectroscopic 3 −ε ∂L v / kT p = (5) N'* = gle k −ε (13) ∂q ∑ v / kT k N gle and where N’*/N is the fraction of molecules in a particular level, gl is the degeneracy, εv is the vibrational energy, ∂H ∂H q = p = − (6,7) k is the Boltzmann constant and T is temperature. For k ∂ k ∂ vibrational energy distribution, there is no degeneracy. pk qk This means that there is only one state per level or gl is In terms of Cartesian coordinates, the equations of unity. motion become However, since vibrational energy increases in discrete increments, a relationship must be formed & & & & between the quantum energy levels and the vibrational = pi = −∇ = energy. This relationship is given by ri pi u fi (8,9) mi ε = ω ()ν + − ω ()ν + 2 − ω + ω These equation of motion must now be integrated to v e .5 xe e .5 .5 e .25xe e (14) obtain the trajectories of the atoms. One of the most common finite difference methods is the velocity Verlet where ν is the quantum energy level, ωe is the spacing algorithm.4 This technique has the advantage of being of the vibrational energy levels, and xeωe is the first third-order accurate even though it contains no third- anharmonic correction.6 order derivatives. The velocity Verlet is a two step In contrast, the molecular dynamics code outputs process. First, the position is advanced one time step vibrational energy which must then be converted to and the velocity is advanced half of a time step. discrete quantum levels by solving equation 14 for ν. This operation is performed on every molecule at every ∆t 2 ν r(t + ∆t )= r ()t + v ()t ∆t + a()t (10) time step. One further complication is that must be an 2 integer. The most efficient method to convert ν is ∆t ∆t simply to truncate it. Truncation ensures that the zero vt + = v()t + a()t (11) energy level bin is equal in size to the other bins. The 2 2 last step is to count the number of molecules in each energy level. At this point the results can be compared At this point, the forces are computed and velocity is to kinetic theory. advanced another half time step. Reactions ∆t ∆t v()t + ∆t = vt + + a()t + ∆t (12) 2 2 The second part of this investigation focuses on modeling simple chemical reactions using molecular This method offers simplicity and good stability for dynamics. The first step is to simulate the dissociation a relatively large time step. The velocity Verlet and recombination of a single molecular species, in this algorithm also has the advantage that it is relatively case, hydrogen and oxygen.