# Presentation: 471955

##  Material Properties at High Energy Density          from Quantum Molecular Dynamics 

**Symposium on Recent Advances in Plasma Physics**
**  ****--- Ron Davidson Celebration**
**PPPL, Princeton, June 10-12, 2007**

**Michael Desjarlais**
**Science Campaigns/HEDP Theory**
**Sandia National Laboratories**

## Sandia’s Z Machine is used for several HEDP experimental campaigns

**Z pinches for Inertial Confinement Fusion**
**Magnetically launched flyer plates for EOS studies (Hugoniots)**
**Ramp wave compression for EOS studies ****(isentropes)**

**The Z machine delivers more than 20 million Amperes of current ZR will generate more than 25 MA**

## HEDP computer simulations rely on “physics packages”:  Conductivities, Equations of State, and Opacities

- This highly structured portion of phase space is Warm Dense Matter

- Definitions of Warm Dense Matter are varied, but generally center around **strongly coupled ions and moderately degenerate electrons** --- many different interactions are comparable.
- A **quantum mechanical** treatment is generally necessary.
- Most wide-range equations of state interpolate through this difficult area.

## Modifications of the Lee-More conductivity algorithms were made to obtain an improved wide-range model*

**Copper at 6000 K**

- (LMD)

- * M. P. Desjarlais, Contrib. Plasma Phys. **41 **(2001) 2-3, 267-270

**Data from Alan DeSilva’s exploding wire experiments**

## A demanding application:  Ultra-high velocity magnetically launched flyer plates  (30 km/sec,  Multi-Mbar pressures)

- Anode/Flyer Plate

- Target

## We simulated these magnetically launched flyer plates using the modified Lee-More (LMD) conductivities

**Detailed comparison between simulations and experiments for magnetically launched flyer plates suggested that our *****improved***** conductivities were still not sufficiently accurate for the warm dense liquid aluminum.**

**For many of our applications we require conductivities accurate to well within a factor of two.**

**Simulations by Ray Lemke with Sandia’s 3D Rad-MHD ALEGRA code**

## "On two occasions I have been asked [by members of Parliament], 

- "On two occasions I have been asked [by members of Parliament],
- 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?'  I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question."
-  - Charles Babbage  (1791-1871)

## Density Functional Theory (DFT) is a formally exact representation of the N electron Schrödinger Equation

**Density Functional Theory (DFT) is a formally exact representation of the N electron Schrödinger Equation**

- “Easy” problem to solve               ( scales like N3 or better)

- Hard problem to solve (scales like N5 )

- Hohenberg and Kohn proved this (1964)
- Nobel prize in Chemistry for Kohn in 1998

## We are using Density Functional Theory (DFT) to perform Quantum Molecular Dynamics (QMD) simulations of Warm Dense Matter

- The simulations are performed with VASP (Vienna Ab initio Simulation Program), a plane wave density functional code (with in-house modifications for our HEDP applications)
- Exchange and Correlation functionals are Generalized Gradient Approximation (GGA/PBE)
- We typically use up to 256 atoms, but it varies depending on density and the number of electrons we need to carry (the valence);  We use Projector Augmented Wave (PAW) all-electron, frozen core potentials for the atoms
- We generally perform our simulations in the Canonical Ensemble (N,V,T) using either velocity scaling or a Nosé-Hoover thermostat to regulate the temperature; Fermi statistics for the electrons
- Typical runs cover one to twenty picoseconds

## The Kubo-Greenwood formula is used to calculate the frequency dependent electrical conductivity

- where *e* and *m* are the electron charge and mass.  The *i* and *j* summations are over the *N* discrete bands of the triply periodic calculation for the cubic supercell with volume . The coordinate index is ** and in general we average over **  to improve the statistics.  *F(**** **i,****k****)* is the Fermi weight corresponding to the energy  for the *I-*th band at ***k*** with wavefunction *i*,**k**.

- We integrate over the Brillouin zone using the method of special **k**-points

- ,

- and average over 10 to 20 configurations selected from the MD run.

**	****Other optical properties are obtained through the ****Kramers-Krönig relations for **

## At liquid aluminum densities just below solid, the optical conductivity is well fit by the Drude model

**The agreement with the Drude model indicates ‘nearly free’ electrons**

**Ion cores displayed with iso-surfaces of the mean valence charge density**

## At lower density, where phase separation ispronounced, a gap begins to form at low energy

**The dc conductivity has dropped by a factor of 25 for a factor of 4 drop in density**

**Note the pronounced separation into liquid and void (vapor) regions**

## At low densities and higher temperatures, a Drudecomponent reemerges in the optical conductivity

**The peak at about 5.8 eV is characteristic of a 3s->3p transition of the isolated ion.**

**The optical conductivity for 0.025 g/cc at 30000 K**

- We have gone from condensed matter         to an ideal plasma!

## The QMD-KG results are in good agreement with Alan DeSilva’s exploding wire data

- [Desjarlais, Kress, and Collins, PRE **66, **025401 (2002)]

## The calculated liquid aluminum conductivities are higher than the improved Lee-More (LMD) model predictions

**The dashed line shows the 2000 K isotherm from our QMD-tuned wide-range aluminum model**

- Dense solid
- ( 5 g/cc )

- Liquid

- Melt

- LMD model

## Flyer plate simulations with the QMD based conductivities give very good agreement with experiment

**Conductivities based on the QMD calculations have given us a new predictive capability**

**Simulations by Ray Lemke with Sandia’s 3D Rad-MHD ALEGRA code**

## The absorption has the proper Hagen-Rubens low frequency behavior, consistent with the dc conductivity

- 

- Hagen-Rubens

- 4*s **** *4*p*

- 3*d **** *4*p*

- Free electrons (Drude Z ~ 0.6)

**We are using these results to improve our low energy opacities**

## We are using quantum molecular dynamics simulations with density functional theory to build manifestly consistent conductivity, EOS, and opacity models

**We are using quantum molecular dynamics simulations with density functional theory to build ****manifestly consistent**** conductivity, EOS, and opacity models**

- Stainless Steel (Fe, Cr, Ni)

**Notes:**
This is an animation.
    We start with a QMD simulation (using density functional theory) at a given density and temperature. The figure shows the Fe, Cr, and Ni atoms (in atomic rations of 38:11:5)
 along with an iso-surface of the electron density.  From this molecular dynamics simulation we extract the Energy and Pressure from time averages (typically over 1 to 2 picoseconds).
These quantities (over a suitable grid) can be used to generate and equation of state.  In this case we used fits to a soft sphere model in the warm dense matter regime and blended
with SESAME at high temperatures and densities.  (Thermodynamic consistency was enforced through thermodynamic integration of the pressure.)
   For selected snapshots (10 or 20), we extract the eigenvalues and wavefunctions and use the Kubo-Greenwood formula to compute the optical conductivity for each snapshot.
These are averaged to produce an average optical conductivity.   The optical conductivity gives us the dc conductivity in the zero frequency limit, and, through the application of
the Kramers-Kronig relations, the low energy (< approx 50 eV) opacities and other optical properties (reflectivities).  In this manner we have generated the electrical conductivity model,
the equation of state, and the low energy opacities on a manifestly consistent basis.   
The final figure shows a color coded conductivity phase space (Log scale ! ) with some special lines or features from the equation of state.  Also shown is a snapshot of an exploding stainless steel
wire projected into this phase space (black triangles).   The snapshot is just at the instant when the voltage on the wire is beginning to collapse as a consequence of the large conducting
 corona that has formed around the wire (note the low density, high temperature, and high conductivity component).   This figure illustrates the complexity of phase space we need to accurately model in order to perform truly representative simulations.   With these advances we have had unprecedented success in modeling magnetically launched flyer plates for our Z  EOS work,  as well as modeling single exploding wires (Cornell aluminum and Sandia stainless steel).

## Recent and active research areas

**Principal Hugoniot and reshock properties of deuterium**
**Principal Hugoniot and Isentrope of aluminum, release isentropes**
**QMD based conductivity models for Al, W, Be, and stainless steel**
**Shock melting of Be and diamond (National Ignition Campaign)**
**Electrical and thermodynamic properties of water at high energy densities**
**Liquid-vapor critical points for Al, W, and stainless steel**
**Research on advanced electronic structure methods for HEDP **
**    ****(finite temperature Exact Exchange, finite temperature GW)**

**Special thanks to my close collaborators:**** Marcus Knudson and Thomas Mattsson**