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-12-0053 August 19, 2014

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

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

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

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

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

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

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

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 2 1−2𝛽 𝑓 𝑛 +2𝛽 𝑓 𝑛+1 𝑝 𝑛+1 = 𝑝 𝑛 +Δ𝑡 1−𝛾 𝑓 𝑛 +𝛾 𝑓 𝑛+1

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.

Implementation Leveraging existing resources: HotQC LAMMPS + Reax/C

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).

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, 1040-1053 (2008).

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