theory and simulation of warm dense matter targets*

The Heavy Ion Fusion Virtual National Laboratory

Theory and Simulation of Warm Dense Matter Targets*

J. J. Barnard1, J. Armijo2, R. M. More2, A. Friedman1, I. Kaganovich3, B. G. Logan2, M. M. Marinak1, G. E. Penn2,

A. B. Sefkow3, P. Santhanam2, P.Stoltz4, S. Veitzer4, J .S. Wurtele2

16th International Symposium on Heavy Ion Inertial Fusion

9-14 July, 2006

Saint-Malo, France1. LLNL 2. LBNL 3. PPPL 4. Tech-X

*Work performed under the auspices of the U.S. Department of Energy under University of California contract W-7405-ENG-48 at LLNL, University of California contract DE-AC03-76SF00098 at LBNL, and and contract DEFG0295ER40919 at PPPL.


Outline of talk

1. Motivation for planar foils (with normal ion beam incidence) approach to studying WDM (viz a viz GSI approach of cylindrical targets or planar targets with parallel incidence)

2. Requirements on accelerators

3. Theory and simulations of planar targets -- Foams

-- Solids-- Exploration of two-phase regime

Existence of temperature/density “plateau”Maxwell construction

-- Parameter studies of more realistic targets-- Droplets and bubbles


Strategy: maximize uniformity and the efficient use of beam energy by placing center of foil at Bragg peak

Ion beam

In simplest example, target is a foil of solid or metallic foam

Enter foilExit foil

dE/dX T

Example: Neon beamEentrance=1.0 MeV/amuEpeak= 0.6 MeV/amuEexit = 0.4 MeV/amudE/dX)/(dE/dX) ≈ 0.05

€

−1

Z 2

dE

dX

Energyloss rate

Energy/Ion mass

(MeV/mg cm2)

(MeV/amu)

(dEdX figure from L.C Northcliffeand R.F.Schilling, Nuclear Data Tables,A7, 233 (1970))

uniformity and fractional energy loss can be high if operate at Bragg peak (Larry Grisham, PPPL)


Various ion masses and energies have been considered for Bragg-peak heating

Beam parameters needed to create a 10 eV plasmain 10% solid aluminum foam, for various ions(10 eV is equivalent to ~ 1011 J/m3 in 10% solid aluminum)

As ion mass increases, so does ion energy and accelerator cost

As ion mass increases, current decreases. Low mass requires neutralization

~ cs= const

Beam A Energy dE/dX Foil Delta z Beam t_hydro= Beam BeamIon at at Entrance for 5% Energy /(2 )z cs Power current

Bragg Bragg Energy variation =10for T eV 10 at eV per 1 for mmPeak Peak ( )approx (10% solid per . sq mm diameter

)Al sq mm spot( )amu ( )MeV ( - / 2)MeV mg cm ( )MeV ( )microns ( / 2)J mm ( ) ns / 2GW mm ( )A

Li 6.94 1.9 2.05 2.8 34.3 5.1 0.8 6.1 1706.2Na 22.99 15.9 11 23.9 53.5 8.0 1.3 6.1 200.3K 39.10 45.6 18.6 68.4 90.8 13.6 2.2 6.1 69.8Rb 85.47 158 39.1 237.0 149.7 22.4 3.7 6.1 20.2Cs 132.91 304 59.2 456.0 190.2 28.5 4.7 6.1 10.5

Initial Hydra simulations confirm temperature uniformity of targets at 0.1 and 0.01 times solid density of Aluminum

0.1 solid Al

0.01solid Al(at t=2.2 ns)

z = 48

r =1 mm

z = 480

Axi

s o

f sy

mm

etry

(simulations are for 0.3 C, 20 MeV Ne beam -- possible NDCX II parameters).

0 1 mmr

time (ns)0

eV

0.7

2.0

1.2

1.0

eV2.2


Metallic foams ease the requirement on pulse duration

tpulse << thydro = z/cs

With foams easier to satisfy

thomogeneity~ n rpore/cs where n is a number of order 3 - 5, rpore is the pore size and cs is the sound speed.

Thus, for n=4, rpore=100 nm, z =40 micron (for a 10% aluminum foam foil):

thydro/thomogeneity ~ 100

But foams locally non-uniform. Timescale to become homogeneous

But need to explore solid density material as well!


The hydrodynamics of heated foils can range from simple through complex

Uniform temperature foil, instantaneously heated, ideal gas equation of state

Uniform temperature foil, instantaneously heated,realistic equation of state

Foil heated nonuniformly, non-instantaneoslyrealistic equation of state

Foil heated nonuniformly, non-instantaneoslyrealistic equation of state, and microscopic physics of droplets and bubbles resolved

Most idealized

Most realistic

The goal: use the measurable experimental quantities (v(z,t), T(z,t), (z,t), P(z,t))to invert the problem: what is the equation of state, if we know the hydro?

In particular, what are the "good" quantities to measure?


The problem of a heated foil may be found in Landau and Lifshitz, Fluid Mechanics textbook (due to Riemann)

At t=0, ,T=0,T0

= const

€

∂∂t

+∂ρv

∂z= 0

∂v

∂t+ v

∂v

∂z= −

1

ρ

∂p

∂z

p= K (adiabatic ideal gas)

(continuity)

(momentum)

Similarity solution can be found for simple waves (cs

2 P/):

z=0

€

v

cs0

=2

γ +1

⎛

⎝ ⎜

⎞

⎠ ⎟

z

cs0t−1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

cs

cs0

=γ −1

γ +1

⎛

⎝ ⎜

⎞

⎠ ⎟

z

cs0t

⎛

⎝ ⎜

⎞

⎠ ⎟+

2

γ +1;

ρ

ρ 0

=cs

cs0

⎛

⎝ ⎜

⎞

⎠ ⎟

2 /(γ −1)

; T

T0

=cs

cs0

⎛

⎝ ⎜

⎞

⎠ ⎟

2

v/cs0

cs/cs0 /0

€

v =−2

γ −1cs0 =

(=5/3shown)

z/(cs0 t)


Simple wave solution good until waves meet at center; complex wave solution is also found in LL textbook

(=z/L)

(= cs0t/L)

0 1

1non-simple waves

simplewaves

Unperturbed

Unperturbed

Using method of characteristicsLL give boundary between simple and complex waves:

€

= −2

(γ −1)τ +

γ +1

γ −1

⎛

⎝ ⎜

⎞

⎠ ⎟τ

3−γ

γ +1

where ζ ≡ z /Land τ ≡ cs0t /L

Unperturbed

simplewaves

2

(for 1)

original foil


LL solution: snapshots of density and velocity

0

v/cs0

0

v/cs0

v/cs0

0 0

v/cs0

( = 5/3)0 =0.5

=1 =2

(Half of space shown, cs0tL )

Time evolution of central T, , and cs for between 53 and 1513

=5/3

=15/13

=5/3

=15/13

=5/3

=15/13

=cs0 t/L

/0

=cs0 t/L

=cs0 t/L

c/cs0T/T0

=15/13

=5/3


Simulation codes needed to go beyond analytic results

In this work 2 codes were used:

DPC: 1D EOS based on tabulated energy levels, Saha equation, melt point, latent heatTailored to Warm Dense Matter regime

Maxwell construction Ref: R. More, H. Yoneda and H. Morikami, JQSRT 99, 409 (2006).

HYDRA: 1, 2, or 3DEOS based on:

QEOS: Thomas-Fermi average atom e-, Cowan model ions and Non-maxwell construction

LEOS: numerical tables from SESAME Maxwell or non-maxwell construction options

Ref: M. M. Marinak, G. D. Kerbel, N. A. Gentile, O. Jones, D.Munro, S. Pollaine, T. R. Dittrich, and S. W. Haan, Phys. Plasmas 8, 2275 (2001).


When realistic EOS is used in WDM region, transition from liquid to vapor alters simple picture

Expansion into 2-phase region leads to -T plateaus with sharp edges1,2

Example shown here is initialized at T=0.5 or 1.0 eV and shownat 0.5 ns after “heating.”

Density

z()

(g/cm3)

0

8

0-3

Temperature

0

1.2

0-3z()

1More, Kato, Yoneda, 2005, preprint. 2Sokolowski-Tinten et al, PRL 81, 224 (1998)

T(eV)

Initial distribution Exact analytic hydro (using numerical EOS)+++++++++ Numerical hydroDPC code results:


HYDRA simulations show both similarities to and differences with More, Kato, Yoneda simulation of 0.5 and 1.0 eV Sn at 0.5 ns

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

Density Temperature

T0 = 0.5 eVT0 = 0.5 eV

(oscillations at phase transition at 1 eV are physical/numerical problems, triggered by the different EOS physics of matter in the two-phase regime)


are needed to see this picture.QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Density

T0 = 1 eV T0 = 1 eV

Temperature

Propagation distance ofsharp interfaceis in approximateagreement

Densityoscillationlikely causedby ∂P/∂ instabilities,(bubbles anddropletsforming?)

Uses QEOSwith no Maxwell construction


Maxwell construction gives equilibrium equation of state

(Figure from F. Reif, "Fundamentals of statistical and thermal physics")

PressureP

Volume V 1/

Isotherms€

∂P

∂V> 0 instability

V

P

Liquid

Gas

Maxwell constructionreplaces unstable regionwith constant pressure2-phase region where liquid and vapor coexist

van der Waals EOSshown


Differences between HYDRA and More, Kato, Yoneda simulations are likely due to differences in EOS

In two phase region, pressureis independent of average density. Material is a combination of liquid (droplets) and vapor (i.e. bubbles).Microscopic densities are at the extreme ends of the constantpressure segment.

10000

Pre

ssu

re (

J/cm

3 )

10

1000

1

0.1

0.01

100

Hydra used modified QEOS data as one of its options. Negative ∂P/∂ (at fixed T) results in dynamically unstable material.

Density (g/cm3)

T=

3.00 eV2.251.691.260.9470.7090.5310.3980.2980.224

3.00 eV2.572.211.891.621.391.191.020.879

0.646

0.476

0.754

0.554

0.300

0.257

0.221

0.350

0.408

T=

1 eV

1 eV

Critical point


Maxwell construction reduces instability in numerical calculations





LEOS without Maxwell constDensity vs. zat 3 ns

LEOS without Maxwell constTemperature vs. zat 3 ns

T(eV)

0

1

(g/cm3)

2

0

z () 0-20 20

z () 0-20 20



LEOS with Maxwell constDensity vs. zat 3 ns

(g/cm3)

2

0



LEOS with Maxwell constTemperature vs. zat 3 ns

z () 0-20 20

z () 0-20 20All four plots: HYDRA, 3.5 foil, 1 ns, 11 kJ/g deposition in Al target

T(eV)

1

0


Parametric studies

Case study: possible option for NDCX II 2.8 MeV Lithium+ beam

Deposition 20 kJ/g 1 ns pulse length3.5 micron solid Aluminum target

Varied: foil thickness finite pulse duration beam intensity EOS/code

Purpose: gain insight into future experiments

0

0.05

0.1

0.15

0.2

0.25

0.3

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

Deposition (kJ/g)

Peak Pressure (Mbar)

3.5 micron

2.5 micron

2.0 micron

1.0 micron

Variations in foil thickness and energy deposition



HYDRA results using QEOSDPC results



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

Deposition (kJ/g)

Maximum Central Temp (eV)

3.5 micron

2.5 micron

2.0 micron

1.0 micron

Pre

ssu

re (

Mb

ar)

Pre

ssu

re (

Mb

ar)

Tem

per

atu

re (

eV)

Tem

per

atu

re (

eV)

Deposition (kJ/g) Deposition (kJ/g)

0

0.5

1

1.5

2

2.5

0.00 10.00 20.00 30.00 40.00

Deposition (kJ/gm)

Avg. Edge Speed (cm/us)

3.5 micron

2.5 micron

2.0 micron

1.0 micron

Expansion velocity is closely correlated with energy deposition but also depends on EOS



Returning to model found in LL:

€

0 =cs0

2

γ(γ −1)

€

v =−2cs0

γ −1

€

⇒

€

v =4γ

γ −1ε0

1/ 2

DPC results HYDRA results using QEOS

In instanteneous heating/perfect gas model outward expansion velocity depends only on 0 and

Su

rfac

e ex

pan

sio

n v

elo

city

Deposition (kJ/g)Deposition (kJ/g)

(cm

/s)


Pulse duration scaling shows similar trends

HYDRA results using QEOS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

Energy Deposited (kJ/g)

Pressure (MBar)

0.5 ns

1 ns

2ns

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

Energy deposited (kJ/g)

Avg. Expansion Velocity (cm/us)

0.5 ns

1 ns

2 ns



DPC results



Pre

ssu

re (

MB

ar)

Pre

ssu

re (

MB

ar)

Exp

ansi

on

sp

eed

(cm

/s)

Exp

ansi

on

sp

eed

(cm

/us)

Energy deposited (kJ/g) Energy deposited (kJ/g)


Expansion of foil is expected to first produce bubbles then droplets

Foil is first entirely liquid then enters 2-phase region

Liquid and gas must be separated by surfaces

Ref: J. Armijo, master's internship report, ENS, Paris, 2006, (in preparation)

Density (g/cm3)

Tem

per

atu

re (

eV)

gas

2-phase

liquid

Example of evolution of foil in and T

DPC result

0 ns

0.2 ns

0.4 ns

0.6 ns0.8 ns

1 ns

Vgas= Vliquid

Maximum size of a droplet in a diverging flow

dF/dx= v(x)

Steady-state droplet

Locally, dv/dx = const (Hubble flow)

x

Ref: J. Armijo, master's internship report, ENS, Paris, 2006, (in preparation)

Equilibrium between stretching viscous force and restoring surface tension

Capillary number Ca= viscous/surface ~ dv/dx x dx / ( x) ~ ( dv/dx x2) / ( x) ~ 1

Maximum size :

Kinetic gas: = 1/3 m v* n l mean free path : l = 1/ 2 n 0

= m v / 3 2 0 Estimate : xmax ~ 0.20 m

AND/OR: Equilibrium between disruptive dynamic pressure and restoring surface tension: Weber number We= inertial/surface ~ ( v2 A )/ x ~ (dv/dx)2 x4 / x ~ 1

Maximum size :

Estimate : xmax ~ 0.05 m x = ( / (dv/dx)2 )1/3

x = / ( dv/dx)

Vel

oci

ty (

cm/s

) 1e6

-1e6Position ()

40-40


Conclusion

We have begun to analyze and simulate planar targets forWarm Dense Matter experiments.

Useful insight is being obtained from:

The similarity solution found in Landau and Lifshitz forideal equation of state and initially uniform temperature

More realistic equations of state (including phase transition from liquid, to two-phase regime)

Inclusion of finite pulse duration and non-uniform energydeposition

Comparisons of simulations with and without Maxwell-construction of the EOS

Work has begun on understanding the role of droplets and bubbles in the target hydrodynamics


Some numbers for max droplet size calculation

Xmax(Ca) = / ( dv/dx) = 100 / 5 10-3 * 109 = 2 10-4 = 0.200

Xmax(We)= ( / (dv/dx)2 )1/3 = (100 / 1* (109)2 )1/3 = 10-16/3 = 10-5.3 cm = 0.050

Surface tension of Al

-200

0

200

400

600

800

1000

0 2000 4000 6000 8000 10000 12000

T (º C)

( / )dyn cm

/Al Ar/Al vacuum

( / )Linear Al vacuum

Critical Point

zone of interest

From CRC Handbook in chemistry and physics

(1964)

Typical conditions for droplet formation: t=1.6ns, T=1eV, liq=1 g/cm3

Surface tension: = 100 dyn/cmThermal speed: v= 500 000 cm/s

Viscosity: = m v / 3 2 0 = 27 * 1.67 10-24 * 5 105 / 3 2 10-16 = 5 10-3 g/(cm-s)Velocity gradient: dv/dx= 106 cm/s / 10-3 cm = 109 s-1

T

theory and simulation of warm dense matter targets*

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