Theory and Simulation of Warm Dense Matter Targets

1
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-ENG-48 at LLNL,
University of California contract
DE-AC03-76SF00098 at LBNL, and and contract
DEFG0295ER40919 at PPPL.
2
Outline of talk

• 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)
• 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

3
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


uniformity and fractional energy loss can be high
if operate at Bragg peak (Larry Grisham, PPPL)
Ion beam
Energy loss rate
DdE/dX ? DT
Example Neon beam Eentrance1.0 MeV/amu Epeak
0.6 MeV/amu Eexit 0.4 MeV/amu (DdE/dX)/(dE/dX)
0.05
(MeV/mg cm2)
Enter foil
Exit foil
(dEdX figure from L.C Northcliffe and
R.F.Schilling, Nuclear Data Tables, A7, 233
(1970))
Energy/Ion mass
(MeV/amu)
4
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)
r e cs const

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

As ion mass increases, so does ion energy and
accelerator cost
5
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
1.0
Axis of symmetry
1.2
2.0
0
r
1 mm
eV
Dz 480 m
2.2
(simulations are for 0.3 mC, 20 MeV Ne beam --
possible NDCX II parameters).
6
Metallic foams ease the requirement on pulse
duration
With foams easier to satisfy
Dtpulse ltlt 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 n4, rpore100 nm, Dz 40 micron (for a
10 aluminum foam foil)
thydro/thomogeneity 100

But need to explore solid density material as
well!
7
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-instantaneosly realistic
equation of state Foil heated nonuniformly,
non-instantaneosly realistic 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), 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?
8
The problem of a heated foil may be found in
Landau and Lifshitz, Fluid Mechanics textbook
(due to Riemann)
At t0, r,Tr0,T0 const
(continuity)
(momentum)
p Krg (adiabatic ideal gas)
z0
Similarity solution can be found for simple
waves (cs2 ? gP/r)
cs/cs0
r/r0
v/cs0
(g5/3 shown)
z/(cs0 t)
9
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
(for tgt1)
10
LL solution snapshots of density and velocity
(Half of space shown, tcs0t/L )
(g 5/3)
t0
t0.5
r/r0
r/r0
v/cs0
v/cs0
t1
t2
r/r0
r/r0
v/cs0
v/cs0
11
Time evolution of central T, r, and cs for g
between 5/3 and 15/13
g15/13
g15/13
g5/3
g5/3
T/T0
c/cs0
tcs0 t/L
tcs0 t/L
g5/3
g5/3
g15/13
g15/13
12
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).
13
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)
Numerical hydro

DPC code results
8
1.2
Temperature

• (g/cm3)

T (eV)
Density
0
0
0
-3
0
-3
z(m)
z(m)
Example shown here is initialized at T0.5 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)
14
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)
Propagation distance of sharp interface is in
approximate agreement
Density
Temperature
T0 0.5 eV
T0 0.5 eV
Uses QEOS with no Maxwell construction
Density oscillation likely caused by ?P/?r
instabilities, (bubbles and droplets forming?)
Temperature
Density
T0 1 eV
T0 1 eV
15
Maxwell construction gives equilibrium equation
of state
P
van der Waals EOS shown
Pressure P
V
Liquid
? instability
Gas
Maxwell construction replaces unstable
region with constant pressure 2-phase region
where liquid and vapor coexist
Isotherms
Volume V ?1/r
(Figure from F. Reif, "Fundamentals of
statistical and thermal physics")
16
Differences between HYDRA and More, Kato, Yoneda
simulations are likely due to differences in EOS
Critical point
10000
3.00 eV 2.57 2.21 1.89 1.62 1.39 1.19 1.02 0.879
3.00 eV 2.25 1.69 1.26 0.947 0.709 0.531 0.398 0.2
98 0.224
T
1000
T
100
Pressure (J/cm3)
10
1 eV
1
1 eV
0.1
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.
17
Maxwell construction reduces instability in
numerical calculations
2
2
LEOS without Maxwell const Density vs. z at 3 ns
LEOS with Maxwell const Density vs. z at 3 ns
r (g/cm3)
r (g/cm3)
0
0
1
1
LEOS with Maxwell const Temperature vs. z at 3 ns
LEOS without Maxwell const Temperature vs. z at 3
ns
T (eV)
T (eV)
0
0
All four plots HYDRA, 3.5 m foil, 1 ns, 11 kJ/g
deposition in Al target
18
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
19
Variations in foil thickness and energy deposition
HYDRA results using QEOS
DPC results
Pressure (Mbar)
Pressure (Mbar)
Temperature (eV)
Temperature (eV)
Deposition (kJ/g)
Deposition (kJ/g)
20
Expansion velocity is closely correlated with
energy deposition but also depends on EOS
DPC results
HYDRA results using QEOS
Surface expansion velocity
(cm/s)
Deposition (kJ/g)
Deposition (kJ/g)
Returning to model found in LL
In instanteneous heating/perfect gas model
outward expansion velocity depends only on e0
and g
21
Pulse duration scaling shows similar trends
DPC results
Pressure (MBar)
Pressure (MBar)
Expansion speed (cm/us)
Expansion speed (cm/s)
Energy deposited (kJ/g)
Energy deposited (kJ/g)
22
Expansion of foil is expected to first produce
bubbles then droplets
Example of evolution of foil in r and T
DPC result
1 ns
0.8 ns
0.6 ns
0.4 ns
Temperature (eV)
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)
23
Maximum size of a droplet in a diverging flow
1e6

• dF/dx ? v(x)

Locally, dv/dx const (Hubble flow)
Steady-state droplet
Velocity (cm/s)
s
x
-1e6
40
-40
Position (m)

• 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 (r v2 A )/? x ?
  (dv/dx)2 x4 /? x 1
• ? Maximum size
• ? Estimate xmax 0.05 ?m

x ? / (? dv/dx)
x (? / ? (dv/dx)2 )1/3
Ref J. Armijo, master's internship report, ENS,
Paris, 2006, (in preparation)
24
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 Comp
arisons 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
25
Some numbers for max droplet size calculation
From CRC Handbook in chemistry and physics (1964)
r
T
Typical conditions for droplet formation
t1.6ns, T1eV, ?liq1 g/cm3 Surface tension
? 100 dyn/cm Thermal 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

• Xmax(Ca) ? / (? dv/dx) 100 / 5 10-3 109 2
  10-4 0.200 m
• Xmax(We) (? / ? (dv/dx)2 )1/3 (100 / 1 (109)2
  )1/3 10-16/3 10-5.3 cm 0.050 m