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Theory and Simulation of Warm Dense Matter Targets*

Theory and Simulation of Warm Dense Matter Targets*.

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Theory and Simulation of Warm Dense Matter Targets*

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  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 Eentrance=1.0 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. 0.1 solid Al eV 0.01 solid Al (at t=2.2 ns) 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 << thydro =Dz/cs But foams locally non-uniform. Timescale to become homogeneous thomogeneity~ n rpore/cs wherenis a number of order 3 - 5, rporeis the pore size andcsis 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!

  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 t=0, r,T=r0,T0 = const (continuity) (momentum) p= Krg (adiabatic ideal gas) z=0 Similarity solution can be found for simple waves (cs2gP/r): cs/cs0 r/r0 v/cs0 (g=5/3 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 t>1)

  10. LL solution: snapshots of density and velocity (Half of space shown, t=cs0t/L ) (g = 5/3) t=0 t=0.5 r/r0 r/r0 v/cs0 v/cs0 t=1 t=2 r/r0 r/r0 v/cs0 v/cs0

  11. r/r0 t=cs0 t/L Time evolution of central T, r, and cs for gbetween 5/3 and 15/13 g=15/13 g=15/13 g=5/3 g=5/3 T/T0 c/cs0 t=cs0 t/L t=cs0 t/L g=5/3 g=5/3 g=15/13 g=15/13

  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 -3 0 z(m) z(m) 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)

  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. 0.754 0.646 0.554 0.476 0.408 0.350 0.300 0.257 0.221 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 T= 3.00 eV 2.25 1.69 1.26 0.947 0.709 0.531 0.398 0.298 0.224 1000 T= 100 10 Pressure (J/cm3) 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. z (m) z (m) z (m) z (m) -20 -20 -20 -20 0 0 0 0 20 20 20 20 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 LEOSwith 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 Locally, dv/dx = const (Hubble flow) • dF/dx=  v(x) Steady-state droplet Velocity (cm/s) s x -1e6 40 -40 Position (m) x =  / ( dv/dx) • 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)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 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

  25. Some numbers for max droplet size calculation From CRC Handbook in chemistry and physics (1964) r T Typical conditions for droplet formation: t=1.6ns, T=1eV, liq=1 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

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