The Heavy Ion Fusion Virtual National Laboratory Theory and Simulation of Warm Dense Matter Targets* J. J. Barnard 1, J. Armijo 2, R. M. More 2, A. Friedman 1, I. Kaganovich 3, B. G. Logan 2, M. M. Marinak 1, G. E. Penn 2, A. B. Sefkow 3, P. Santhanam 2, P.Stoltz 4, S. Veitzer 4, J.S. Wurtele 2 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.

The Heavy Ion Fusion Virtual National Laboratory 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

The Heavy Ion Fusion Virtual National Laboratory 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 foil Exit foil  dE/dX   T Example: Neon beam E entrance =1.0 MeV/amu E peak = 0.6 MeV/amu E exit = 0.4 MeV/amu  dE/dX)/(dE/dX) ≈ 0.05 Energy loss rate Energy/Ion mass (MeV/mg cm 2 ) (MeV/amu) (dEdX figure from L.C Northcliffe and 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)

The Heavy Ion Fusion Virtual National Laboratory 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 ~ J/m 3 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 ~  c s = const

Initial Hydra simulations confirm temperature uniformity of targets at 0.1 and 0.01 times solid density of Aluminum 0.1 solid Al 0.01 solid Al (at t=2.2 ns)  z = 48  r =1 mm  z = 480  Axis of symmetry (simulations are for 0.3  C, 20 MeV Ne beam -- possible NDCX II parameters). 0 1 mm r time (ns) 0 eV eV 2.2

The Heavy Ion Fusion Virtual National Laboratory Metallic foams ease the requirement on pulse duration  t pulse << t hydro =  z/c s With foams easier to satisfy t homogeneity ~ n r pore /c s where n is a number of order 3 - 5, r pore is the pore size and c s is the sound speed. Thus, for n=4, r pore =100 nm,  z =40 micron ( for a 10% aluminum foam foil): t hydro /t homogeneity ~ 100 But foams locally non-uniform. Timescale to become homogeneous But need to explore solid density material as well!

The Heavy Ion Fusion Virtual National Laboratory 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),  (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 Heavy Ion Fusion Virtual National Laboratory The problem of a heated foil may be found in Landau and Lifshitz, Fluid Mechanics textbook (due to Riemann) At t=0, ,T=  ,T 0 = const p= K   (adiabatic ideal gas) (continuity) (momentum) Similarity solution can be found for simple waves (c s 2   P/  ): z=0 v/c s0 c s /c s0 /0/0 (  =5/3 shown) z/(c s0 t)

The Heavy Ion Fusion Virtual National Laboratory Simple wave solution good until waves meet at center; complex wave solution is also found in LL textbook  (=z/L)  (= c s0 t/L) 01 1 non-simple waves simple waves Unperturbed Using method of characteristics LL give boundary between simple and complex waves: Unperturbed simple waves 2 (for  1) original foil

The Heavy Ion Fusion Virtual National Laboratory LL solution: snapshots of density and velocity   v/c s0   v/c s0   v/c s0 (  = 5/3)  0  =0.5  =1  =2 (Half of space shown,  c s0 t  L )

Time evolution of central T, , and c s for  between  and   =5/3  =15/13  =5/3  =15/13  =5/3  =15/13  =c s0 t/L /0/0 c/c s0 T/T 0  =15/13  =5/3

The Heavy Ion Fusion Virtual National Laboratory 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).

The Heavy Ion Fusion Virtual National Laboratory 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 edges 1,2 Example shown here is initialized at T=0.5 or 1.0 eV and shown at 0.5 ns after “heating.” Density z(  )  (g/cm 3 ) Temperature z(  ) 1 More, Kato, Yoneda, 2005, preprint. 2 Sokolowski-Tinten et al, PRL 81, 224 (1998) T (eV) Initial distribution Exact analytic hydro (using numerical EOS) Numerical hydro DPC code results:

The Heavy Ion Fusion Virtual National Laboratory 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 Density Temperature T 0 = 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) Density T 0 = 1 eV Temperature Propagation distance of sharp interface is in approximate agreement Density oscillation likely caused by ∂P/∂  instabilities, (bubbles and droplets forming?) Uses QEOS with no Maxwell construction

The Heavy Ion Fusion Virtual National Laboratory Maxwell construction gives equilibrium equation of state (Figure from F. Reif, "Fundamentals of statistical and thermal physics") Pressure P Volume V  1/  Isotherms  instability V P Liquid Gas Maxwell construction replaces unstable region with constant pressure 2-phase region where liquid and vapor coexist van der Waals EOS shown

The Heavy Ion Fusion Virtual National Laboratory Differences between HYDRA and More, Kato, Yoneda simulations are likely due to differences in EOS 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 Pressure (J/cm 3 ) Hydra used modified QEOS data as one of its options. Negative ∂P/∂  (at fixed T) results in dynamically unstable material. Density (g/cm 3 ) T= 3.00 eV eV T= 1 eV Critical point

The Heavy Ion Fusion Virtual National Laboratory Maxwell construction reduces instability in numerical calculations LEOS without Maxwell const Density vs. z at 3 ns LEOS without Maxwell const Temperature vs. z at 3 ns T (eV) 0 1  (g/cm 3 ) 2 0 z (  ) z (  ) LEOS with Maxwell const Density vs. z at 3 ns  (g/cm 3 ) 2 0 LEOS with Maxwell const Temperature vs. z at 3 ns z (  ) z (  ) All four plots: HYDRA, 3.5  foil, 1 ns, 11 kJ/g deposition in Al target T (eV) 1 0

The Heavy Ion Fusion Virtual National Laboratory 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

Variations in foil thickness and energy deposition HYDRA results using QEOS DPC results Pressure (Mbar) Temperature (eV) Deposition (kJ/g)

Expansion velocity is closely correlated with energy deposition but also depends on EOS Returning to model found in LL: DPC results HYDRA results using QEOS In instanteneous heating/perfect gas model outward expansion velocity depends only on   and  Surface expansion velocity Deposition (kJ/g) (cm/s)

The Heavy Ion Fusion Virtual National Laboratory Pulse duration scaling shows similar trends DPC results Pressure (MBar) Expansion speed (cm/s) Expansion speed (cm/us) Energy deposited (kJ/g)

The Heavy Ion Fusion Virtual National Laboratory 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/cm 3 ) Temperature (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 ns 0.8 ns 1 ns V gas = V liquid

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 x 2 ) / (  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 : x max ~ 0.20  m  AND/OR: Equilibrium between disruptive dynamic pressure and restoring surface tension: Weber number We= inertial/surface ~ (  v 2 A )/  x ~  (dv/dx) 2 x 4 /  x ~ 1  Maximum size :  Estimate : x max ~ 0.05  m x = (  /  (dv/dx) 2 ) 1/3 x =  / (  dv/dx) Velocity (cm/s) 1e6 -1e6 Position (  )

The Heavy Ion Fusion Virtual National Laboratory 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 Maxwell- construction of the EOS Work has begun on understanding the role of droplets and bubbles in the target hydrodynamics

The Heavy Ion Fusion Virtual National Laboratory Some numbers for max droplet size calculation X max (Ca) =  / (  dv/dx) = 100 / * 10 9 = =  X max (We)= (  /  (dv/dx) 2 ) 1/3 = (100 / 1* ( 10 9 ) 2 ) 1/3 = /3 = cm =  From CRC Handbook in chemistry and physics (1964) Typical conditions for droplet formation: t=1.6ns, T=1eV,  liq =1 g/cm 3 Surface tension:  = 100 dyn/cm Thermal speed: v= cm/s Viscosity:  = m v / 3  2  0 = 27 * * / 3  = g/(cm-s) Velocity gradient: dv/dx= 10 6 cm/s / cm = 10 9 s -1 T 