1. Hierarchical Theoretical Methods for Understanding and Predicting Anisotropic Thermal Transport and Energy Release in Rocket Propellant Formulations
Michael Ortiz
California Institute of Technology
Univeristy of Missouri PI: Thomas D. Sewell
Subcontract: EC-SRP
August 19, 2014

2. Objectives Perform long-term atomistic modeling of chemical reactions
Observe influence of anisotropy in heat conduction on combustion rates

3. Test Case: Combustion of Graphite
Input: Graphite orientation
Output: Reaction-front speed
Oxygen
Graphite
Reaction Front

4. Test Case 1. O2 reservoir: reaction front 3. Far field:
2. Reaction zone:
Full atomistics (Reax)
Nonequil. stat. mech.
Mass/heat transport
1. O2 reservoir:
Coarse-grained
atomistics (QC)
Lagrangian gas
solver
3. Far field:
Lagrangian solid
reaction front
The point here is that graphite is a model material in that its transverse thermal conductivity is two orders of magnitude smaller than its in-plane thermal conductivity

5. Methods Employ a number of methods to reduce computation effort:
Maximum-Entropy
Atomic heat transport
Implicit mesoscopic dynamics
Quasi-continuum

6. Maximum Entropy Optimization
Input: Probability density
𝜌 𝑞 , 𝑝
Objective function: Entropy
𝑆 𝜌 =− 𝑘 𝐵 ∫𝜌 𝑞 , 𝑝 log 𝜌 𝑞 , 𝑝 𝑑 𝑞 𝑑 𝑝
Constraints: Known variance
𝑒 𝑖 = Γ 𝜌 𝑞 , 𝑝 ℎ 𝑖 𝑞 , 𝑝 𝑑 𝑞 𝑑 𝑝
Solution: Optimal probability density
𝜌 ∗ = argmax 𝜌 𝑒 𝑖 = ℎ 𝑖 𝑆 𝜌

7. Maximum-Entropy
By solving for the probability density that maximizes entropy within a given class of functions, we obtain a modified potential as a function of temperature
This modified potential accounts for thermal vibrations statistically and behaves more smoothly than the underlying potential
Thus, simulations can proceed with long time steps

8. Atomic Heat Transport
The temperature of each atom evolves according to a discrete heat equation:
𝑑 𝑑𝑡 1 𝑘 𝐵 𝜕Φ 𝜕 𝛽 𝑖 𝛽 = 𝑗≠𝑖 𝜕𝜓 𝜕 𝑃 𝑖𝑗 𝑘 𝐵 𝛽 𝑖 − 𝛽 𝑗

9. Implicit Mesoscale Dynamics
Employ a Newmark time-stepping algorithm to update mean positions and momenta
𝑚 𝑞 𝑛+1 = 𝑚 𝑞 𝑛 +Δ𝑡 𝑝 𝑛 +Δ 𝑡 −2𝛽 𝑓 𝑛 +2𝛽 𝑓 𝑛+1
𝑝 𝑛+1 = 𝑝 𝑛 +Δ𝑡 1−𝛾 𝑓 𝑛 +𝛾 𝑓 𝑛+1

10. Quasi-continuum Coarse-grain space adaptively:
Full atomistic resolution within reaction zone
Continuum approximation away from reaction zone
The point is that we want both spatial coarse-graining away from the reaction zone (achieved using the quasicontinuum method) in addition to temporal coarse-graining (achieve by NESM)
Tadmor, E. B., Phillips, R., & Ortiz, M. (1996). Mixed Atomistic and Continuum Models of Deformation in, 7463(3), 4529–4534.

11. Implementation
Leveraging existing resources:
HotQC
LAMMPS + Reax/C

12. HotQC
HotQC is a code written by M. Ponga for simulating nanovoid growth in Cu
We are collaborating with him to repurpose HotQC to handle multiple species (i.e. C and O) and the Reax potential
Ponga, M. (2013). Multiscale modeling of point defects evolution at finite temperature : nanovoids and vacancies, (January).

13. LAMMPS + Reax/C Implementation of Reax potential in LAMMPS
Calculate energy and forces as a function of:
Atom positions
Atom species
Parameters are available for C, H, O, and N interaction
Can be called as a library
Plimpton, S. (1995). Fast Parallel Algorithms for Short – Range Molecular Dynamics, 117(June 1994), 1–42.
Chenoweth, van Duin and Goddard, Journal of Physical Chemistry A, 112, (2008).

14. Progress Implemented Future work Low-temperature or max-ent mechanics
Full atomistic model representation
Explicit dynamics model updates
Lennard-Jones or Reax potential
Small simulation domain
Quasi-continuum model representation
Implicit dynamics model updates
Heat transport
Large simulation domain