Transcript Document 7727979

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, France
1. 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-ENG48 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
In simplest example, target is a foil of solid or metallic foam
Ion beam
Energy
loss rate
1 dE
 2
Z dX
(MeV/mg cm2)
Enter foil
Exit foil
Energy/Ion mass
(MeV/amu)
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uniformity and
fractional energy loss
can be high if operate
at Bragg peak (Larry
Grisham, PPPL)
DdE/dX  DT
Example: Neon beam
Eentrance=1.0 MeV/amu
Epeak= 0.6 MeV/amu
Eexit = 0.4 MeV/amu
(DdE/dX)/(dE/dX) ≈ 0.05
(dEdX figure from L.C Northcliffe
and R.F.Schilling, Nuclear Data Tables,
A7, 233 (1970))
Various ion masses and energies have been
considered for Bragg-peak heating
Beam parameters needed to create a 10 eV plasma
in 10% solid aluminum foam, for various ions
(10 eV is equivalent to ~ 1011 J/m3 in 10% solid aluminum)
Beam
Ion
Li
Na
K
Rb
Cs
A
(amu)
6.94
22.99
39.10
85.47
132.91
Energy
at
Bragg
Peak
dE/dX
at
Bragg
Peak
(MeV) (MeV-mg/cm2)
1.9
2.05
15.9
11
45.6
18.6
158
39.1
304
59.2
Foil
Entrance
Energy
(approx)
(MeV)
2.8
23.9
68.4
237.0
456.0
Delta z
Beam
t_hydro=
Beam
Beam
Dz/(2 cs)
for 5%
Energy
Power
current
variation for T=10eV at 10 eV
per
for 1mm
(10% solid
per
sq. mm
diameter
Al)
sq mm
spot
(microns)
(J/mm2)
(ns)
GW/mm2
(A)
34.3
5.1
0.8
6.1
1706.2
53.5
8.0
1.3
6.1
200.3
90.8
13.6
2.2
6.1
69.8
149.7
22.4
3.7
6.1
20.2
190.2
28.5
4.7
6.1
10.5
As ion mass
increases, so does
ion energy and
accelerator cost
~re cs= const
As ion mass
increases, current
decreases.
 Low mass requires
neutralization
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Initial Hydra simulations confirm temperature uniformity of
targets at 0.1 and 0.01 times solid density of Aluminum
time (ns)
0
Dz = 48 m
r =1 mm
0.7
0.1
solid
Al
1.0
eV
1.2
2.0
0
r
1 mm
Dz = 480 m
0.01
solid Al
(at t=2.2 ns)
2.2
(simulations are for 0.3 mC, 20 MeV Ne beam -- possible NDCX II parameters).
eV
Metallic foams ease the requirement on pulse
duration
With foams easier to satisfy
Dtpulse << thydro = Dz/cs
But foams locally non-uniform. Timescale to become homogeneous
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, Dz =40 micron (for a 10% aluminum foam foil):
thydro/thomogeneity
~ 100
But need to explore solid density material as well!
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The hydrodynamics of heated foils can range
from simple through complex
Most idealized
Uniform temperature foil, instantaneously heated,
ideal gas equation of state
Uniform temperature foil, instantaneously heated,
realistic equation of state
Foil heated nonuniformly, non-instantaneosly
realistic equation of state
Most realistic
Foil heated nonuniformly, non-instantaneosly
realistic equation of state, and microscopic
physics of droplets and bubbles resolved
The goal: use the measurable experimental quantities (v(z,t), T(z,t), r(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)
r rv
(continuity)

0
t z
v
v
1 p
v 
(momentum)
t
z
r z
p= Krg(adiabatic ideal gas)
Similarity solution can be found
for simple waves (cs2  gP/r):
At t=0,
r,T=r0,T0
= const
z=0
cs/cs0
r/ r0

v  2  z
 
1

c s0 g  1c s0 t 
c s g 1 z  2
 
;
 
c s0 g  1c s0 t  g  1
2 /(g 1)
r  c s 
  
r 0 c s0 
v/cs0
T  c s 
  
T0 c s0 
2
;
v
2
c 
g 1 s0
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
z/(cs0 t)
(g=5/3
shown)
Simple wave solution good until waves meet at center;
complex wave solution is also found in LL textbook
t (= cs0t/L)
non-simple waves
1
simple
waves
simple
waves
Unperturbed
Unperturbed
Unperturbed
0
1
2
z (=z/L)
original foil
Using method of characteristics
LL give boundary between simple and complex waves:
3g


2
g  1 g 1
z
t  
t
(for t>1)
(g 1)
g 1
where z  z /L and t  c s0t /L
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LL solution: snapshots of density and velocity
(g = 5/3)
(Half of space shown, tcs0t/L )
t0
r/r0
t=0.5
r/r0
v/cs0
t=1
r/r0
v/cs0
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v/cs0
t=2
r/r0
v/cs0
Time evolution of central T, r, and cs for g
between 5/3and 15/13
g=15/13
c/cs0
g=5/3
g=15/13
T/T0
t=cs0 t/L
t=cs0 t/L
r/r0
g=5/3
g=5/3
g=5/3
g=15/13
g=15/13
t=cs0 t/L
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 heat
Tailored to Warm Dense Matter regime
Maxwell construction
Ref: R. More, H. Yoneda and H. Morikami, JQSRT 99, 409 (2006).
HYDRA:
1, 2, or 3D
EOS 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 r-T plateaus with sharp edges1,2
Initial distribution
Exact analytic hydro (using numerical EOS)
+++++++++
DPC code results:
Numerical hydro
8
1.2
r
(g/cm3)
T
(eV)
Density
0
-3
0
z(m)
Temperature
0
-3
z(m)
0
Example shown here is initialized at T=0.5 or 1.0 eV and shown
at 0.5 ns after “heating.”
1More,
Kato, Yoneda, 2005, preprint. 2Sokolowski-Tinten et al, PRL 81, 224 (1998)
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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
(oscillations at phase transition at 1 eV are physical/numerical problems, triggered by the different
EOS physics of matter in the two-phase regime)
Density
T0 = 0.5 eV
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Density
oscillation
likely caused
by ∂P/∂r
instabilities,
(bubbles and
droplets
forming?)
Density
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
T0 = 1 eV
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Temperature
T0 = 0.5 eV
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Temperature
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
T0 = 1 eV
Propagation
distance of
sharp interface
is in
approximate
agreement
Uses QEOS
with no
Maxwell
construction
Maxwell construction gives equilibrium
equation of state
van der Waals EOS
shown
Pressure
P
P
V
P
> 0  instability
V
Liquid
Gas
Maxwell construction
replaces unstable region

Isotherms
with constant pressure
2-phase region where
liquid and vapor coexist
Volume V 1/r
(Figure from F. Reif, "Fundamentals of statistical and thermal physics")
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Differences between HYDRA and More, Kato, Yoneda
simulations are likely due to differences in EOS
Critical point
10000
0.754
0.646
0.554
0.476
0.408
0.350
0.300
1 eV
1000
Pressure (J/cm3)
T=
3.00 eV
2.57
2.21
1.89
1.62
1.39
1.19
1.02
0.879
T=
100
10
1 eV
1
0.257
3.00 eV
2.25
1.69
1.26
0.947
0.709
0.531
0.398
0.298
0.224
0.1
0.221
0.01
Density (g/cm3)
In two phase region, pressure
is 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 constant
pressure segment.
Hydra used modified QEOS data as one
of its options. Negative ∂P/∂r (at fixed
T) results in dynamically unstable
material.
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Maxwell construction reduces instability in
numerical calculations
2
r
(g/cm3)
0
1
T
(eV)
2
LEOS without Maxwell const
Density vs. z
at 3 ns
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
z (m)
-20
0
r
(g/cm3)
TIFF (LZW) decompressor
are needed to see this picture.
0
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
0
20
LEOS without Maxwell const
Temperature vs. z
at 3 ns
QuickTime™ and a
LEOS with Maxwell const
Density vs. z
at 3 ns
z (m)
1
T
(eV)
-20
0
20
LEOS with Maxwell const
Temperature vs. z
at 3 ns
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
0
z (m)
-20
0
20
z (m)
All four plots: HYDRA, 3.5 m foil, 1 ns, 11 kJ/g deposition in Al target
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-20
0
20
Parametric studies
Case study: possible option for NDCX II
2.8 MeV Lithium+ beam
Deposition 20 kJ/g
1 ns pulse length
3.5 micron solid Aluminum target
Varied: foil thickness
finite pulse duration
beam intensity
EOS/code
Purpose: gain insight into future experiments
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Variations in foil thickness and energy
deposition
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
0.3
(Mbar)
Pressure
Peak Pressure
(Mbar)
Pressure (Mbar)
DPC results
HYDRA results using QEOS
0.25
0.2
3.5 micron
2.5 micron
0.15
2.0 micron
1.0 micron
0.1
0.05
0
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
(eV)
Temperature
Maximum Central Temp (eV)
Temperature (eV)
Deposition (kJ/g)
2
1.8
1.6
1.4
2.5 micron
1
2.0 micron
0.8
1.0 micron
0.6
0.4
0.2
0
0.00
Deposition (kJ/g)
3.5 micron
1.2
5.00
10.00
15.00
20.00
25.00
Deposition (kJ/g)
Deposition
(kJ/g)
30.00
35.00
2.5
DPC results
HYDRA results using QEOS
2
Avg. Edge Speed (cm/us)
Surface expansion velocity
(cm/s)
Expansion velocity is closely correlated with
energy deposition but also depends on EOS
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
1.5
3.5 micron
2.5 micron
2.0 micron
1.0 micron
1
0.5
0
0.00
10.00
20.00
30.00
40.00
Deposition (kJ/gm)
Deposition (kJ/g)
Deposition (kJ/g)
Returning to model found in LL:
2
c s0
e0 
g (g 1)
2c s0
v
g 1

4 g 1/ 2
v
e0
g 1
In instanteneous heating/perfect gas model outward expansion velocity
depends only on e0 and g


Pulse duration scaling shows similar trends
HYDRA results using QEOS
0.6
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Energy deposited (kJ/g)
Pressure (MBar)
0.5
0.4
0.5 ns
0.3
1 ns
2ns
0.2
0.1
Avg. Expansion Velocity (cm/us)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Expansion speed (cm/us) Pressure (MBar)
Expansion speed (cm/s)
Pressure (MBar)
DPC results
0
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
Energy Deposited (kJ/g)
1.8
1.6
1.4
1.2
0.5 ns
1
1 ns
0.8
2 ns
0.6
0.4
0.2
0
0.00
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5.00
10.00
15.00
20.00
25.00
Energy deposited (kJ/g)
30.00
35.00
Energy deposited (kJ/g)
40.00
Expansion of foil is expected to first produce
bubbles then droplets
Temperature (eV)
Example of evolution of foil in rand T
1 ns
0.8 ns
0.6 ns
DPC result
0.4 ns
gas
liquid
0.2 ns
2-phase
Vgas= Vliquid
Density (g/cm3)
0 ns
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)
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Velocity (cm/s)
Maximum size of a droplet in a diverging flow
1e6
Locally, dv/dx
= const
(Hubble flow)
dF/dx= m v(x)
x
s
Steady-state
droplet
-1e6
-40
40
Position (m)
 Equilibrium between stretching viscous force and restoring surface tension
Capillary number Ca= viscous/surface ~  m dv/dx x dx / (s x) ~ (m dv/dx x2) / (s x) ~ 1
 Maximum size :
x = s / (m dv/dx)
Kinetic gas: m = 1/3 m v* n l
 m= m v / 3 2 s0
mean free path : l = 1/ 2 n s0
 Estimate : xmax ~ 0.20 mm
 AND/OR: Equilibrium between disruptive dynamic pressure and restoring surface
tension: Weber number We= inertial/surface ~ (rv2 A )/s x ~ r (dv/dx)2 x4 /s x ~ 1
 Maximum size :
x = (s / r (dv/dx)2 )1/3
 Estimate : xmax ~ 0.05 mm
Ref: J. Armijo, master's internship report, ENS, Paris, 2006, (in preparation)
Conclusion
We have begun to analyze and simulate planar targets for
Warm Dense Matter experiments.
Useful insight is being obtained from:
The similarity solution found in Landau and Lifshitz for
ideal 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 energy
deposition
Comparisons of simulations with and without Maxwellconstruction 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
r
Surface tension of Al
From CRC Handbook in chemistry and physics (1964)
1000
Al/Ar
Al/vacuum
Linear (Al/vacuum)
800
s (dyn/cm)
600
400
zone of interest
200
Critical Point
0
0
T
2000
4000
6000
8000
10000
12000
-200
T (º C)
Typical conditions for droplet formation:
t=1.6ns, T=1eV, rliq=1 g/cm3
Surface tension: s = 100 dyn/cm
Thermal speed: v= 500 000 cm/s
Viscosity: m= m v / 3 2 s0 = 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
Xmax(Ca) = s / (m dv/dx) = 100 / 5 10-3 * 109 = 2 10-4 = 0.200 m
Xmax(We)= (s / r (dv/dx)2 )1/3 = (100 / 1* (109)2 )1/3 = 10-16/3 = 10-5.3 cm = 0.050 m
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