May 5-6, 2003/ARR 1

1. Liquid Wall Ablation under IFE Photon EnergyDeposition

2. Flibe Properties from EoS

A. René Raffray and Mofreh ZaghloulUniversity of California, San Diego

ARIES Town Meeting on Liquid Wall Chamber DynamicsHilton Garden Inn, Livermore, CA

May 5-6, 2003

May 5-6, 2003/ARR 2

Physical Processes in X-Ray Ablation

EnergyDeposition &

Transient HeatTransport

InducedThermal- Spikes

MechanicalResponse

PhaseTransitions

•Stresses and Strainsand HydrodynamicMotion•Fractures and Spall

• Surface Vaporization•Heterogeneous Nucleation•Homogeneous Nucleation(Phase Explosion)

Ablation Processes

Expansion,Cooling and

Condensation

SurfaceVaporization

Phase ExplosionLiquid/Vapor

Mixture

Spall Fractures

Liquid

FilmX-Rays

Fast Ions

Slow Ions

Impulse

Impulse

sy

sx

sz

May 5-6, 2003/ARR 3

High Photon Heating Rate Could Lead to Explosive Boiling

Photon-likeheating rate

Ion-likeheatingrate

• Effect of free surface vaporization isreduced for very high for heating rate(photon-like)

• Vaporization into heterogeneous nucleiis also very low for high heating rate

From K.Song and X.Xu, AppliedSurfaceScience 127-129 (1998)111-116

• Rapid boiling involving homogeneousnucleation leads to superheating to ametastable liquid state

• The metastable liquid has an excessfree energy, so it decomposesexplosively into liquid and vaporphases.

- As T/Ttc increases past 0.9, Becker-Döhring theory of nucleation indicate anavalanche-like and explosive growth of nucleation rate (by 20-30 orders of magnitude)

May 5-6, 2003/ARR 4

Analysis Based on Photon Spectra for 458 MJ IndirectDrive Target

• High photon energy for indirect drive target case (25% compared to 6% ion energy))• More details on target spectra available on ARIES Web site:

http://aries.ucsd.edu/ARIES/

(25%)

(1%)

May 5-6, 2003/ARR 5

Photon Energy Deposition Density Profile in Flibe Filmand Explosive Boiling Region (for Rchamber=6.5m)

1x107

1x108

1x109

1x1010

1x1011

1x1012

0 5 10 15

Ener

gy d

epos

ition

(J/m

3 )

Penetration depth (micron)

Cohesion energy (total evaporation energy)

2.5

Evap.region

10.4

2-phase region

Sensible energy (energy to reach saturation)

Sensibleenergy basedon uniformvaporpressurefollowingphotonpassage inchamber andincludingevaporatedFlibe fromfilm

0.9 Tcritical

4.1

Explo.boil.region

Bounding estimates of thermally-induced aerosol source term from photon energy deposition:(1)Upper bound: the whole 2-phase region; (2)Lower bound: explosive boiling region

May 5-6, 2003/ARR 6

1x106

1x107

1x108

1x109

1x1010

1x1011

1x1012

1x1013

0 1 2 3 4 5

Ener

gy d

epos

ition

(J/m

3 )

Penetration depth (micron)

Photon Energy Deposition Density Profile in Pb Film and Explosive Boiling Region (for Rchamber=6.5m)

Cohesion energy (total evaporation energy)

Sensible energy (energy to reach saturation)

0.9 Tcritical

1.9 2.5 3.9

2-phase region

Explo.boil.region

Evaporated region

May 5-6, 2003/ARR 7

Explosive Boiling Results for Rchamber=3.5 m

Explosive Boiling Ablated Thickness for:Flibe Pb

Rchamber= 6.5 m 4.1 mm 2.5 mm Rchamber= 3.5 m 10.9 mm 3.83 mm

May 5-6, 2003/ARR 8

Mechanical Response to Induced Shock

• Rapid increase in internal energy due to x-ray energy deposition andablation impulse creates high pressure within the material

- The induced shock wave propagates to the wall, gets reflected back to the freesurface where a rarefaction wave is produced (creating tensile stresses) andpropagates back through the material. The process is repeated.

- If the magnitude of the rarefaction wave is greater than the tensile strength ofthe material, fracture or spall will occur establishing a new surface.

• Evolution of spall in a body subject to transient stresses is complex

- Material dependent: brittle, ductile or liquid

- Small perturbations can lead to opening of voids and initiation of spall process

- A reasonable prediction of the dynamic spall strength, time to failure, and some measure of the nominal fragment size created in the spall event are needed to characterize the spall process

- An upper bound theoretical spall strength can be derived from intermolecularpotential

May 5-6, 2003/ARR 9

Theoretical (Maximum) Spall Strength Provides an UpperBound Estimate in the Absence of Appropriate Spall Data

˙˚

˘ÍÎ

Ș¯

ˆÁË

Ê ---˜

¯

ˆÁË

Ê --=

a)vv(

exp2a

)vv(2expU)v(U 00

coh

UUcoh= Specific cohesive energy

v = 1/r = Specific volume

v0 = Specific volume at zero pressure

a=(2v0 Ucoh / B0 )1/2

B0 = Bulk modulus

˙˚

˘ÍÎ

Ș¯

ˆÁË

Ê ---˜

¯

ˆÁË

Ê --=-=

a)vv(

expa

)vv(2exp

aU2

dvdU

)v(P 00coh

- The cold pressure is given by:- The cold pressure is given by:

TheoreticalTheoretical spall spall strength,strength, P Pthth,given by minimum of P(v):

†

Pth =Ucoh B0

8 v0

• Based on intermolecular potential reflecting dependence on cohesive energy and bulk modulus with inherent energy balance

- Using a three-parameter potential such as the Morse potential

May 5-6, 2003/ARR 10

Spall Strength is Highly Temperature Dependent

• Using the Soft Sphere EOS to Estimate Temperature-Dependent Spall Strength

˙˙˚

˘

ÍÍÎ

Ș̃¯

ˆÁÁË

Ê-˜̃

¯

ˆÁÁË

Ê++˜̃

¯

ˆÁÁË

Ê++=

TKTKQ)4n(

61

TKC

23

TKNU)T,v(UB

m

3/1

B

9/n

B

3/nnBcoh

er

er

er

˙˙˚

˘

ÍÍÎ

Ș̃¯

ˆÁÁË

Ê-˜̃

¯

ˆÁÁË

Ê++˜̃

¯

ˆÁÁË

Ê+=

TKm

TKQ)4n(n

181

TKCn

31

1V

TKN)T,v(P

B

m

3/1

B

9/n

B

3/nn

B er

er

er

• Theoretical spall strength is then calculated from:

.0vd

)T,v(Pd=

n, m, Q, e, and s are adjustable parametersto satisfy the available experimental data

N: Number of molecules,

V: Specific volume,

r= N s3 / (21/2) V,

s : the sphere diameter,

Cn: FCC Madelung constant.

May 5-6, 2003/ARR 11

Temperature-Dependent Spall Strengths of Example Materials

-0.0657Gas-0.22353749

-0.2814-0.1010-0.52212999-0.6848-0.4267-0.89812250-1.4212-0.8950-1.40981450-2.4914-1.4401-2.0014750

Flibe (GPa)Li (GPa)Pb (GPa)T (K)

May 5-6, 2003/ARR 12

Parameters of Different IFE Reactor Design Studies &Pressure Pulse Profile

17.3

0.008

6.5

23.31

0.04

3.5

15.2

0.026

6.5

115

Pb/SiC

3.553.55Closest distance from target(m)

45.77

0.022

115

Flibe/SS

Present Study

6059.022Reactive impulse** (Pa s)

0.02340.02780.0125Vaporized mass (kg/m2)

95129*31X-ray yield (MJ)

Pb-17Li/SiCFlibe/CPb/SiCLiquid/Structure

HiballOsirusPrometheus-LParameters

* X-ray and debris

• The pressure wave is steady (nochange in shape)

• Parameters of different IFEreactor design studies and thepresent study are comparable

• Osirus profile scaled according tothe relative impulses and used forthe present analysis.

** Ablated material velocity ~ sonic velocity ~ 586/2094 m/s for Pb/Flibe at Tcrit (~5100/4500 K)

May 5-6, 2003/ARR 13

Illustration of Spalling Following a Pressure WavePropagation in the Flibe Layer on a Perfectly Stiff Wall

r @ 2000 kg/m3, Cs @ 3300 m/s, Tin = 885.7 K, Pth = -1.887 GPa

Spalled Thickness @ 2.1 µm & Spall Time (t3 – t1) @ 16.9 ns

Spall time from the beginning of the pressure pulse = 2 L/Cs+ (t3-t1)@ 200 ns for a 0.3 mm flibe layer

1. For a perfectly stiff wall,the pressure wave isreflected from the wall andreturns to the free surfaceas a pressure pulse

2. Pfree-surface= Pchamber and thepressure pulse arriving atthe free boundary must bereflected back as a tensilewave

3. If the net tensile stress >the spall strength of thematerial, rupture occursestablishing a newsurface

May 5-6, 2003/ARR 14

Summary for the Case of Perfectly Stiff Wall

3.66.2R=6.5 m

5.615.0R=3.5 mTotal fragmentedthickness

(µm)

1.12.1R=6.5 m

1.84.1R=3.5 mSpallation thickness(µm)

2.54.1R=6.5 m

3.810.9R=3.5 mExplosive boilingthickness

(µm)

PbFlibe

May 5-6, 2003/ARR 15

BUT No Wall is Perfectly Stiff !

†

P = G P0 =(Zwall - Z film )(Zwall + Z film )

P0

6.09C46.7SS5.75SiC

19.31Pb6.84Flibe

Z (Mkg/m2 s)MaterialA pressure pulse with initial pressure P0 will be reflected atthe mismatched interface with a new pressure, P where:

With the acoustic impedance Z = r Cs

In principle, combination ofliquid and structural materialcan be chosen to eliminate orminimize spalling

E.g. For Pb/SiC and 3.5 m Cavity Radius fi G= -0.541• Pressure wave is reflected as a reduced tensile wave.• The max. pres., Pmax, of the reflected wave = -6.67 GPa• Spall tensile strength, Pth, = -1.875 GPa

• ÁPmaxÁ>ÁPthÁ fi Spall occurs (at 0.94 µm from the wall)

May 5-6, 2003/ARR 16

1.784

3.585

6.8 > |Pth|

18.2 > |Pth|

0.745(reflected as a

reduced pressurepulse)

0.745(reflected as a

reduced pressurepulse )

Flibe/SS

-0.541(reflected as areduced tensile

pulse)

-0.058(reflected as areduced tensile

pulse)

R = 3.5m

G-0.541

(reflected as areduced tensile

pulse)

-0.058(reflected as areduced tensile

pulse)

R = 6.5m

L-0.240.0R = 6.5m

L-0.960.0R = 3.5mSpallation

thickness(µm)

4.35 > |Pth|0.53< |Pth|R = 6.5m

6.67 > |Pth|1.4 < |Pth|R = 3.5m

Magnitude ofthe maximum

reflectedpressure

Pmax(GPa)

Pb/SiCFlibe/C

Spallation Analysis for Different ProposedLiquid/Wall Combinations

May 5-6, 2003/ARR 17

Summary of Study of Wall Ablation Mechanisms asAerosol Source Term

• Integrated effect of liquid wall thermal and mechanical responsesto X-ray energy deposition to provide bounding estimates ofablation as source term for aerosol analysis- First principle consideration

- Ablation depths of liquid wall from explosive boiling and spalling

- Spall strength of materials as compared to anticipated IFE shocks

- Acoustic impedances of structural material and liquid can be chosen to avoid or minimize spalling

• Some key remaining questions- Form (vapor conditions, size and distribution of liquid droplets) of the

removed material?

- What happens to the spalled material (not enough energy to push it out…where does it go?)

- Spall time scale

May 5-6, 2003/ARR 18

Flibe Properties from EoS

M. ZaghloulUniversity of California, San Diego

ARIES Town Meeting on Liquid Wall Chamber DynamicsHilton Garden Inn, Livermore, CA

May 5-6, 2003

May 5-6, 2003/ARR 19

Flibe Thermodynamic Properties

Liquid+

Vapor

Vaporchemical

decomposition of molecules

Plasmamultiple

ionization

TTcritcrit ~ 4500 K~ 4500 K

T ~ 1 T ~ 1 eVeV

Tem

pera

ture

Te m

pera

ture

ß The soft-sphere equation of state for liquid flibetakes into account the dependence on r and T(*).

ß Vapor pressure correlations from ORNL & Olanderet al., Fusion Technology,41, 141 (2002).

ß Chemical equilibrium codes that use data fromJANAF tables are used (e.g. the computer codeSTANJAN, Stanford Univ.)

ß Fitted equations of state for flibe gas(**).

ß Inaccuracies include

- Use of Post-Jensen ionization data calculated from theCorona equilibrium (valid only for extremely low densities and very high temperatures-optically thin plasma-and show no dependence on density).

- Post-Jensen data give no information on the individualpopulations needed for the internal energy computationand so further approximations have to be made.

- No Coulomb corrections.

- No excitation energies included in the computations ofinternal energies.

(*)(*) Xiang Xiang M. Chen M. Chen , Virgil E. Schrock, and Per F. Peterson ““ The soft-sphere equation of state for liquid The soft-sphere equation of state for liquid Flibe Flibe,,”” Fusion Tech. 21, 1525 (1992). Fusion Tech. 21, 1525 (1992).

(**) Xiang M. Chen, Virgil E. Schrock, and Per F. Peterson, “Fitted Equation of State for Flibe Gas,” Fusion Technology 26, 912 (1994).

Improved Modeling andComputations in the High-Temperature Region are Needed

(Revisited)

May 5-6, 2003/ARR 20

Flibe Vapor Pressure and Stability of LiBeF3(*)

(*) Mofreh Zaghloul, D. K. Sze, A. R. Raffray, “Thermo-physical Properties and Equilibrium VaporComposition of Lithium-Fluoride Beryllium-Fluoride (LiF/BeF2) Molten Salt ,” 15th Topical Meeting on the

Technology of Fusion Energy, TOFE, Washington, DC, Nov. 2002.

• Based on previous experimental results by Buchler and Stauffer(***), Olanderet al(**) assumed a LiBeF3 contribution of ~10% to flibe vapor, and estimatedthe LiBeF3 vapor pressure based on a negative energy of formation (DG0 < 0)

• However, from Knack et al. and using NIST-JANAF tables, DG0 for LiBeF3 >0 for T > 555 K (Meta-stable)

• Accordingly, Olander et al. overestimated the vapor pressure of LiBeF3 by~ 30 orders of magnitude

• However the effect on their estimation of the overall vapor pressure of flibe ismuch smaller since LiBeF3 contribution was assumed to be ~10%

(**) Olander et al., Fusion Technology,41, 141 (2002).

(***) O. KNACKE, O. KUBASCHEWSKI and K. HESSELMANN (Eds.), Thermochemical Properties of Inorganic Substances I, second edition, Springer-Verlag, Berlin, Germany (1991).

)g(3LiBeF)g(2BeF)g(LiF =+

May 5-6, 2003/ARR 21

Proposed Correlation for Flibe Vapor PressureOver a Wide Range of Temperature

One can combine the valuescomputed for flibe vaporpressure using thermo-chemical data from NIST-JANAF tables in the lowtemperature region toORNL measurements(adequately accurate for T>1000 C) and use a best fitto obtain a correlation validover a wide range of temp.

)K(T/208.11026424.9Plog Torr10 -=

May 5-6, 2003/ARR 22

AssumptionsI- Fully-Dissociated Flibe-Gas (T> 1 eV) (Mixture of Monatomic Gases);

(**) T. Fujimoto et al, Phys. Rev. A 42, 6588 (1990).

(***) W. Ebeling et al, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids (Akademic-Verlag, 1976)

Ionization Equilibrium and Thermodynamic Functions(*)(*)Mofreh R. Zaghloul, “A consistent model for the equilibrium thermodynamic functions of partially

ionized Flibe plasma with Coulomb Corrections,” Physics of Plasmas, 10, No. 2 (2003) 527-538.

˜̃¯

ˆÁÁË

Ê-= Â

£

= TK

EexpgU

B

n,j,rIE

1nn,j,rj,r

effj,rn,j,r

Terminated Partition Function

j,rj,reff

j,r III D-=

Continuum Lowering(***)

V- Coulomb Corrections

F,Be,Lij,ZZi3

1jav

3

1jj,e

max,Z

1ij,i

j

===Â ÂÂ= ==

aIV- Electro-neutrality.

III- No Nuclear Reactions (Constant Number of Heavy Particles ) nH= Const.;

5.1nuc

a

67nuc

e2nuc

183c

)Z/49.0(55.0a

,10Z

TZ105.1)cm(n

-=

˜¯

ˆÁË

Ê¥=-

II- Local Thermodynamic Equilibrium LTE(Validity Criterion by Fujimoto(**))

May 5-6, 2003/ARR 23

Thermodynamic Relations

CoulavBH

NNT

PZTKnV

FP

ke

D++=˜̃¯

ˆÁÁË

Ê

∂

∂-= )1(

},{,

˜˜¯

ˆÁÁË

Ê+˜̃

¯

ˆÁÁË

Ê+-+-+=

˜¯

ˆÁË

涶

-=⋅+==

  Â= = =

Coul

J

1j

Z

1i

i

1zi,jzziH0BavH

}N,N{,V

2

EW)II(n)TT(K)Z1(n231

TF

TT

1)STF(

1E

1e

jmax,

ke

DDar

rrr

1- Pressure

2- Internal energy

3- Contribution ofexcitation energies ofion i of element j tointernal energy ( ) ( )ÂÂ

-£-£

--=

∂∂

=

i,ji,j,i,ji,ji,j,i,j II

B,i,j,i,j

II

B,i,j,i,j,i,j

i,j2

Bi,j

TKexpgTKexpg

UlnT

TKW

Dw

zzz

Dw

zzzz

zz

www

4- Enthalpy )Pe(1

he += rr

5- Adiabatic exponent ,C/C vP=g

2/1

T

2/1

S

s

PPv ˙

˚

˘ÍÎ

Ș̃¯

ˆÁÁË

Ê

∂∂

=˜̃¯

ˆÁÁË

Ê

∂∂

=r

gr6- Sound Speed

The most rigorous expressions for thermodynamic functions have been used

May 5-6, 2003/ARR 24

Results of Detailed Composition & Average Ionization State

• As the temperature increases, the molarfractions of neutral species (Li, Be, and F)decrease monotonically as a result of theprogressive ionization.

• With further increase of the temperature,higher ionized ionic species appear on theexpense of lower-fold ionized species.

• The lower the density the lowerrecombination rate, and the higher <Z>.

• The lower the density the more prominent isthe step-like structure of the <Z>.

• At very high T, the value of <Z> approachesits expected value of a fully-dissociated,fully-ionized flibe-plasma , <Z> = 6.57.

May 5-6, 2003/ARR 25

Results for Flibe Pressure & Internal Energy

• Substantial deviation from the ideal-gas linear behavior.• Ionization and Coulomb interactions among charged particles are responsible

for the deviation in the pressure curves, while for internal energy anothercontribution comes from excitation.

May 5-6, 2003/ARR 26

Results for Flibe Specific Heat, Adiabatic Exponent & Sound Speed

• Non-linearity characteristics of hightemperature plasma are most prominent.

• At very high T, Cp increases to its expectedvalue of a fully-dissociated, fully-ionized flibe-plasma (Cp = 1.1138¥104 J/kg K) and so for theother properties.

May 5-6, 2003/ARR 27

ResultsValidity of LTE Assumption & Effect of Coulomb Corrections

• LTE assumption is applicable fordensities ≥ 0.001 kg/m3 for thetemperature range shown.

• <Z> is the most affected parameter atlow temperatures.

• Percentage relative differences ofother parameters at the same densityare bounded within ± 15 %.

May 5-6, 2003/ARR 28

Comparison with Chen’s et al Results (*).

(*) Xiang M. Chen, Virgil E. Schrock, and Per F. Peterson, Fusion Technology 26, 912 (1994).

• Differences of up to ~ 65% • Differences of up to ~ 150%

May 5-6, 2003/ARR 29

Optical Characteristics of Flibe Multi-Frequency AbsorptionCoefficient

In the evaluation of the multi-frequency absorption coefficient, kn

• Molecular absorption is neglected (the plasma temperature is considered to besufficiently high > 1 eV that the plasma is fully dissociated);

• Photon energies are low enough (< 1MeV) that high-energy absorption processes suchas pair production and nuclear photoabsorption can be ignored;

• Only electronic transitions in the field of the ions are considered (bound-bound, bound-free, and free-free) as these constitute the major absorption processes for mostconditions of interest.