# Presentation: 180050

## The slides that follow were presented at the PSAAP Bidder's Meeting May 16-17, 2006 and represent the ASC Trilab authors and interests as presented in the associated White Paper for this subject area.

- Predictive Science Academic Alliance Program (PSAAP)

## Equation of State and Constitutive Properties

- John Aidun (SNL)
- Christian Mailhiot (LLNL)
- Tommy Sewell (LANL)
- Predictive Science Academic Alliance Program
- Bidders’ Meeting
- Dallas, TX

## Equation of State and Constitutive Properties

- Accurate, physics-based constitutive models and fundamental properties are absolutely essential to development of a truly predictive scientific simulation capability.
- Multiscale, strongly heterogeneous materials under a wide variety of thermodynamic and dynamic loading conditions
- Pristine, aged, and damaged states
- Requirement: quantified accuracy & reliability

## Thermodynamic Properties

- EOS establishes fundamental relationships among thermodynamic variables, provides closure of hydrodynamic equations
- Typically well-defined on atomic scales
- Often directly accessible, in principle, to atomistic simulation *(with caveats)*
- EOS, phase boundaries, re-solidification, transport and mechanical properties
- Wide range of thermodynamic states of interest
- Required as input to mesoscopic and continuum constitutive models

## Engineering Constitutive Properties & Models

- Typically governed by phenomena occurring across multiple length scales
- Often dominated by collective behavior of defects, material interfaces, and evolution of material microstructure
- Spatial resolution of practical applications necessitates use of sub-grid models
- Usually phenomenological in nature

## Predictive HE Detonation Products EOS 

**Goal: Utilize ab initio quantum chemistry to develop accurate HE detonation products EOS.**
**People: M. Sam Shaw, C.J. Tymczak (LANL, T-14)**
**MondoSCF is ab initio linear-scaling DFT quantum chemisty code developed by LANL ASC (Challacombe, et al.) that includes exact exchange (e.g., PBE0 6-31G**).**
**Perform 20-100 calculations for each P,T point where configurations are sampled from classical Monte Carlo simulations (100K) with potentials matched to ab initio functions. **
**Experimental data (*********) have error bars that extend to central curve.  Calculated points (O) have error bars of similar size to the symbols.**
**Corrected sampling of configurations generated from model potentials, correction of potentials**
**Basis set and sample size convergence**
**Development and bounds of cross-potentials**

## HMX Crystal Bulk Modulus

| K (GPa), K’ | Comment | Authors | Publication/year |
| --- | --- | --- | --- |
| 13.5,     9.3
0-7.5 Gpa | Isotherm fit:
Us-Up EOS | Olinger et al. | CEA Conference proceeding, 1978 |
| 12.4*,     10.4
0-27 GPa | Isotherm fit: 3rd-order Birch-Murnaghan EOS | Yoo & Cynn | J. Chem. Phys., 1999 |
| 14±3.5,     7.5 ± 1.9 | Re-analysis of experimental data | Menikoff & Sewell | High Press. Res., 2001 |
| 14.53,  9.57 (P21/c)
16.86,  9.50 (P21/n)
0-7.5 GPa | Isotherm fit: Murnaghan EOS
(Atomistic simulation) | Sorescu et al. | J. Phys. Chem., 1999 |
| 15.1 (Cij)
15.5-16.3,     11.0-12.1 (EOS, 0-10 Gpa) | Isotherm fits and Cij (Atomistic simulation) | Sewell et al. | J. Chem. Phys., 2003 |
| 21.0 ± 1.0,  7.45 ± 0.95
0-5 GPa | Isotherm fit: 3rd-order Birch-Murnaghan EOS | Gump & Peiris | J. Appl. Phys., 2005 |
| 9.9
(from isentropic Cij) | Brillouin zone scattering | Stevens & Eckhardt | J. Chem. Phys., 2005  |

## Prediction of TATB Elastic Tensor

**Objective: Calculate isothermal elastic tensor for TATB crystal as function of temperature, pressure, and crystal purity**
- People:  Tommy Sewell, Dmitry Bedrov (Utah), Rick Gee (LLNL)
** **Force field of Gee et al. (LLNL, ‘04) implemented in Lucretius MD code
-  Minor change to original: ignore intramolecular non-bonded terms
-  Computed structure, elastic tensor for T= 195 - 395 K, P=0-3 GPa
-  4x4x6 super cell (192 molecules)
-  Results are still preliminary
-  Considering effect of randomly distributed porosity (0, 5, 10%)

| Case | KReuss | KVoigt | GRuess | GVoigt |
| --- | --- | --- | --- | --- |
| Pure | 10.2 | 25.7 | 2.8 | 14.7 |
| 10% porosity | 6.1 | 15.9 | 1.4 | 9.3 |

- Snapshot from simulation of
- TATB crystal with 10% porosity
- at T=295 K, P=1 atm.

*Preliminary* predictions of isotropic moduli,
- obtained from the full elastic tensor at T=295 K,
- P=1 atm.  Units are Gpa.
- Reuss uniform stress, Voigt uniform strain

| Case | C11 | C22 | C33 |
| --- | --- | --- | --- |
| Pure | 82.6 | 73.8 | 11.6 |
| 10% porosity | 51.6 | 47.4 | 7.2 |

*Preliminary* predictions of
- C11, C22, C33 for T=295 K,
- P=1 atm.  Units are Gpa.

## Phase Transitions, Melt, and Solidification

- Major effects on material response
- 	material strength, microstructure, chemistry
- Pure elements, alloys, molecular compounds of interest
- Phase boundaries, mechanisms and kinetics of transformation
- Physics of phase transitions at interfaces
- Frustration and metastability
- Atomic- through continuum-scale descriptions

## MD simulation of pressure-driven solidification provide kinetic data

**Novel interatomic potential for Sn developed that reproduces DFT based liquid structure at high pressures**

- Simulations provide the microscopic kinetic coefficient
- Interface velocity as a function of under-cooling / over-pressurization
- Simulations provide the solid-liquid interfacial free energy
- Key parameter in rate of homogeneous nucleation

**S. Foiles, J-P Davis - SNL**

## Mesoscale modeling of solidification connects microscopic parameters to the overall freezing rate

- Parameter free prediction of dendrite
- growth velocity in highly under-cooled
- Ni.  (Bragard et al., 2001)

- Demonstrated connecting atomistic simulations to mesoscale for conventional solidification
- Mesoscale models will provide the overall freezing rate required to correlate measurements with liquid-solid transition
- Approaches
- Nucleation and growth models
- Volume fraction as a function of time/temperature/pressure
- Phase Field model of solidification

**J. J. Hoyt - SNL**

## Optical, Electrical, and Transport Properties

- Often measured experimentally, thus the ability to predict these quantities from first principles and relate them to thermodynamic state is of great importance
- Examples: optical properties of shocked materials, electrical conductivity of materials at high temperature

## Constitutive Model Formulations

- Need for material properties well beyond linear regime (e.g., plasticity or failure)
- Single- as well as multi-material treatments
- Strongly-coupled physics
- Strongly-driven conditions
- Robust, efficient algorithms
- Extensive verification and validation

## PBX-9501 DetonationRalph Menikoff (LANL)

- Model I (r): Menikoff
- Solid: Birch Murnaghan (Sewell, Menikoff, 2004)), Cv = Goddard QC
- Gas: Sesame (Shaw, 2004)
- Reaction: Arrhenius (Henson, 2002)
- Does not do initiation (Pop plot)
- Model II (b): Stewart, Lambert
- Solid: Wide-Ranging EOS (Davis, 2004): Cv = k S
- Gas: WR EOS (Davis, 2004)
- Reaction: Single-step Pn
- Also fits Pop plot data
- Model III (g): Tarver I&G
- Solid: JWL (Tarver), Cv = C
- Gas: JWL (Tarver)
- Reaction: time constant and two step Pn
- Also fits Pop plot data

- CJ

- VN

## HMX Shock Compression Studies

- Jaramillo (LANL), Strachan (Purdue), Sewell (LANL)

**245760 Molecules = 6881280 Atoms (C-H bonds fixed)**
**System Size: 184.27 x 19.27 x 19.47 nm**

- Stationary slab

- Collision with slab, particle velocity = -Up

-  Shock propagates to right with speed = Us-Up

## HMX Local Temperature, Plastic Activity

- Jaramillo, Strachan, Sewell

**u****p****=0.75 km/s**
**t=35 ps**

**u****p****=0.75 km/s**
**t=28 ps**

## Polycrystal and Multi-Material Elasticity and Plasticity

- Even anisotropic single-crystal elasticity is difficult to obtain in practice
- Sensitive stress, temperature, and strain-rate dependencies; also, damage and defects
- Polycrystal response requires homogenization over an appropriate representative volume element (RVE)
- Metals, organics, piezo- and ferro-electrics, random and ordered composites
- Complicated time histories

## PBX Constitutive Modeling(Bardenhagen and Brydon, LANL)

- PBX Constitutive Modeling(Bardenhagen and Brydon, LANL)

-  Continuum constitutive theory predicts Plastic-Bonded Explosive (PBX) bulk response given microscale strain statistics.
-  Simulations used to generate strain statistics
-  PBX Microstructures obtained from microtomography
-  *Smallest sample size (    ) gives accurate bulk properties* (moduli)
-  Statistical Representative Volume Element (RVE) ~80x bigger.

- Strain Statistics

- To appear in *Shock Compression of Condensed Matter - 2005*

## SPaSM: Massive Parallel MD to Characterize Shock Response

- *K. Kadau,T. C. Germann, P.S. Lomdahl, B. L. Holian, *Science* **296** 1681 (2002).
- ** D.H. Kalantar et al., *Phys. Rev. Lett.* **95**, 075502 (2005)

- EAM poly Fe 30 M atoms 256-CPU of Linux Cluster Pink
- Bcc (gray), 10c (blue) 11c (yellow), hcp (red) fcc (green)***,****

- 271 nm  271 nm
-  343 nm, with
- 0.41% porosity  **2,131,656,770**** ****atoms**

- Centrosymmetry parameter to show only hcp (grey) and defect (red) atoms

- Shock-induced transformation mechanism in Fe single crystals predicted by large-scale molecular-dynamics simulations*
- NEMD results confirmed experimentally** by ultrafast (nanosecond) X-ray diffraction  (shocks and high intensity X-rays produced by high energy laser systems OMEGA/Janus/Vulcan)

- The effect of an aging-induced small density (~1%) of voids (vacancy clusters) in a metal on the resulting shock Hugoniot, temperature, plastic response, etc. upon loading as the voids partially collapse.
- EAM Cu, Up = 200 m/s, 17 ps

- ***K. Kadau,T. C. Germann, P.S. Lomdahl, B. L. Holian, R. Albers, to be submitted to *Phys. Rev. Lett*. (2006).
- **** B. Yaakobi et al., *Phys. Rev. Lett.* **95**, 075501 (2005)

## Compression of Real Foam Microstructures(Bardenhagen and Brydon, LANL)

- Derivatives of basic Particle-In-Cell (PIC) Method
- Governing equations solved on grid.
- Particles used to advect solution variables.
- No mesh distortion, large material deformations
- Complex geometries easily discretized; importing data from X-ray tomography
- Material Point Method
- Wide variety of materials models readily incorporated
- Robust contact algorithm ((*Bardenhagen et. al., Comput. Model. Eng. & Sci., 2 (2001), 509*)
- Improved interpolation method: GIMP (Bardenhagen, in press)
- Generate PDF’s and RVE’s for homogenization models
- Incorporate fundamental physics models as available

- Stress field of urethane foam at 10, 50 and 90% compression

- Calculated results of foam compression for shown sample size and 1/8 sized sample

- Typical results for foam compression curve

## Interfacial Phenomena

- Material interfaces often dominate constitutive behavior
- Many materials of interest have huge interfacial areas
- Interfaces can be sharp or diffuse, can serve to localize stress, and can affect material stability/compatibility
- Both physical and chemical processes at interfaces are of interest
- Need for simulation/theory advances on all scales

## Inverse Time-Dependent Ginzburg-Landau on Polymer Blends

- 1/~T0.5

- Inverting TDGL Simulations

- Inverting
- MC Simulations

**Mean Field Behavior**
**As Expected for Polymer Melt**

**Goal:  Fit the TDGL Equation of Motion**
**********t********(R,t) = ********2****(A******** + B ********3**** - ********2********)**

## Homogenized Treatment of Damage, Cracking, and Failure

- Material heterogeneity, localization and damage are inherently multi-scale
- Generally required to treat sub-scale phenomena within engineering-scale simulations
- Need physically-based, quantifiable and measurable connections among atomistic, mesoscopic, and anisotropic continuum-scale simulation models and methods