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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.
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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
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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)
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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
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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!
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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?
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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)
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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
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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
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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).
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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:
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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)
QuickTime™ and aTIFF (LZW) decompressor
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
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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
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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
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Maxwell construction reduces instability in numerical calculations
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
LEOS with Maxwell constDensity vs. zat 3 ns
(g/cm3)
2
0
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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
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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
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
HYDRA results using QEOSDPC results
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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)
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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
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
DPC results
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
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)
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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
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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
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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