Equations of State

Damian Swift

P-24 Plasma Physics

Los Alamos National Laboratory

dswift@lanl.gov http://public.lanl.gov/dswift

1

Outline

• What is the equation of state and why should we care?

• Theoretical basis

• Complete and incomplete EOS

• Predicting the EOS

• Experimental measurements

• Corrections, limits, and other issues

2

Nomenclature for thermodynamics

Mass density ρ and specific volume v: ρ = 1/v.

intensive extensive

p pressure e internal energyT temperature s entropy

Extensive quantities: can integrate over system to find total.

Convention here: use specific quantity (per mass).

3

What is the equation of state?

compression

block of matter

heat

What’s the pressure?

Material-dependent property, relating thermodynamic

potential and its natural parameters, e.g. de = Tds − pdv.EOS is the relation e(s, v) for a material.

Derivatives and other functions of EOS give other quantities:

p, T , sound speed c2, heat capacity cv, ...

Perfect gas EOS:

p = nkBT p = (γ − 1)ρe

4

Uses of the equation of state

Hydrocode / continuum mechanics simulations of impact-type

problems:

Cross-section of armor after perforation by projectile.

5

Uses of the equation of state

Divide components into small cells, assume conditions are

spatially uniform in each. Need material properties for

simulations.

6

Uses of the equation of state

Radiation-driven implosions, e.g. NIF hohlraum-driven fusion

capsule:

Source: J. Lindl, “Inertial confinement fusion,” Springer (1998).

7

Simulation of dynamic loading problems

Initial value problem: given {ρ, ~u, e}(~r, t0) over some region

{~r} ∈ R, what is {ρ, ~u, e}(~r, t > t0)?

Continuum equations (Lagrangian, neglecting heat conduction):

∂ρ(~r, t)

∂t= −ρ(~r, t)div ~u(~r, t)

∂~u(~r, t)

∂t= − 1

ρ(~r, t)grad p(~r, t)

∂e(~r, t)

∂t= −p(~r, t)div ~u(~r, t)

Boundary conditions p(~r, t) and/or ~u(~r, t) for {~r} ∈ dR.

Use EOS p(ρ, e) – derived from e(s, v) – to complete equations

and allow integration to proceed.

8

Structure of the equation of state

temperature

pre

ssu

re

black hole

neutronic matter

solid

gas

liquid

BE

C

quark−gluonplasma

pointcritical

fluid plasma

increasingionisation

mo

lecu

lar d

isso

cia

tion

Details depend on the material composition.

9

‘Moderate’ pressure and temperatureGeophysics, hypervelocity impact, terrestrial explosions, ICF, ...

−GPa (pspall) < p < PPa (Gbar); ∼ 10K < T < MeV; ps < t < Gyr

e.g. Cu:

0 1 2 3 4 5 6 7 8 910

ρ (g/cm3)

0 2000 4000 6000 8000 1000012000

T (K)

0123456789

e (MJ/kg)

solid

liquid

critical point

vapor

liquid/vapor

e(ρ, T ) = ecold(ρ) + eion-thermal(ρ, T ) + eelectron-thermal(ρ, T )

10

Cu phase diagram

SESAME #4:

0 1 2 3 4 5 6 7 8 910

ρ (g/cm3)

0 2000 4000 6000 8000 1000012000

T (K)

0123456789

e (MJ/kg)

solid

liquid

critical point

vapor

liquid/vapor

11

Energy, EOS, and phase diagram

Given e(ρ, T ) – for a given atomic arrangement – can construct

EOS using 2nd law of thermodynamics:

de = Tds − pdv ⇒ s(v, T ) = s(v,0) +∫ T

0

dT ′

T ′∂e(v, T ′)

∂T ′

from which calculate f(ρ, T ) = e(ρ, T ) − Ts(ρ, T ) ⇒ EOS for

this phase.

s(v,0): ‘configurational entropy’ – may vary between phases.

At any p, T , equilibrium phase has lowest Gibbs free energy

g = e − Ts + pv (natural parameters g(T, p) as dg = −sdT + vdp)

12

Phase diagram: Fe

Different crystal structures occur in the solid state:

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000 2500 3000 3500

pres

sure

(G

Pa)

temperature (K)

bccliquid

hcp fcc

bcc

13

Multiphase EOS: Fe

2 4 6 8 10 12 14 16 18 20

ρ (g/cm3)0500

10001500

20002500

30003500

40004500

5000

T (K)

-505

1015202530

e (MJ/kg)

solid

liquid

Source: J.T. Gammel, T-1.

14

Multiphase EOS: Fe

7.58

8.59

9.510

ρ (g/cm3)0

200

400

600

800

1000

T (K)

-0.10

0.10.20.30.40.50.60.70.8

e (MJ/kg)ε

α

γ

Source: J.T. Gammel, T-1.

15

Thermodynamic completeness

Complete EOS: thermodynamic potential expressed in natural

parameters

e(s, v), f(T, v), g(T, p), h(s, p)

Can derive any thermodynamic parameter from a complete

EOS.

Continuum mechanics:

∂ρ/∂t = −ρdiv ~u, ∂~u/∂t = −(1/ρ)grad p, ∂e/∂t = −pdiv ~u

only require p(ρ, e).

Also: easiest to deduce p(ρ, e) from mechanical measurements.

Incomplete EOS: p(ρ, e) with no information about T .

‘SESAME’ EOS: e(ρ, T ) and p(ρ, T ): a complete EOS – but

may be inconsistent.

16

Predicting the EOS

e(ρ, T ) = ecold(ρ) + eion-thermal(ρ, T ) + eelectron-thermal(ρ, T )

– solve for electron states for a given set of ion positions (Dirac orSchrodinger equation).

Electrons are indistinguishable fermions: quantum many-body problem...

17

Quantum many-body problem

Ground state of collective wavefunction Ψ: HΨ = E0Ψ.

Numerical quantum mechanics: based on single-particle

wavefunctions {ψi}. Fermions: antisymmetric with respect to

particle exchange (Pauli exclusion principle), e.g. for 2-particle

wavefunctions Ψ(1,2) = −Ψ(2,1), can be satisfied if

Ψ(1,2) =1√2[ψ1(1)ψ2(2) − ψ2(1)ψ1(2)].

More generally,

Ψ(1,2, ...n) =∑

χε(χ)ψχ(i)(i) = detψi(j)

where ε is a permutation operator – ‘Slater determinant’

approach; in practice prohibitive in computer time.

Differences in the phase of the wavefunctions ⇒ correlation

effects: another complication.

18

Local density approximation

Assume can approximate exchange and correlation by

modifying the potential energy contribution in the Hamiltonian

to include functions of the local electron density

n(~r) =∑

i

ψ†i (~r)ψi(~r).

These functions are then calibrated against detailed exchange

and correlation calculations for simple systems e.g. uniform

electron gas.

Typical accuracy: a few percent in density at p = 0, T = 0.

19

High temperature EOSBelow ∼1 eV: ion-thermal from phonons (Bose-Einstein) andelectron-thermal from band structure around Fermi surface (Fermi-Dirac).

Different ion positions (may be small effect: dominated by ρ); excited

electrons / ionization; band structure consistent with T . Treats plasma

seamlessly. Spherical atom model.

20

Measuring the EOS

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000 2500 3000 3500

pres

sure

(G

Pa)

temperature (K)

bcc

hcp fcc

bcc

liquid

isobar

isen

tro

pe

principalHugoniot

porousHugoniots

10% 20% 30%

Isobaric expansion: exploding wire in pressure vessel.

Isotherm: static presses (special case).

Hugoniot: locus of states accessible by a single shock wave

(can vary its strength).

21

Static high pressure

gasket

diamond

load

pressure calibrante.g. ruby (fluorescence)

sample hydrostaticfluid

Apply load at known temperature; measure pressure, mass density,structure, etc. Diagnostics: x-ray diffraction, optical properties.

p to ∼400 GPa, steady T to 2500 K, higher by laser heating.

22

Steady shock waves

speed us

speed u1

speed u1

undisturbed material(stationary: u = 0)

0

directionof motion

shock wavepiston

shocked material

Rankine-Hugoniot relations (conservation laws):

u2s = v2

0

p − p0

v0 − v, up =

√

[(p − p0)(v0 − v)], e = e0 +1

2(p + p0)(v0 − v)

– 3 equations, 5 unknowns ⇒ measure 2 quantities to determine state on

shock Hugoniot.

23

Shock wave experiments

– ways to launch and measure a steady shock:

• Impact experiments (gas gun, powder gun, electromagnetic

flyer, explosively-driven flyer, laser flyer)

• Detonation-driven shock

• Radiation-driven shock (nuclear explosion, laser hohlraum)

• Laser ablation

Aspect ratio: thin samples to preserve 1D region in center.

24

Impact experiments

targetflyer position

tim

e

targetflyer

shockshock

rarefactions

spallplane

impact

tension

releaseof tension

“Classical”: same material for both; measure uflyer = 2up and

us from transit time.

25

Gas guns

e.g. two-stage light gas gun:

barrel

target / catch tank

pistonsabot, flyer

target

light gas(hydrogen)

powder breech

diagnostics: flyer speed, flatness,shock transit time, surface velocity

26

Electromagnetic launcher

e.g. Z pulsed power machine:

and ablationaccelerate flyerfrom surface

electricalcurrent flow

performed in a recessin the panel wall

each experiment is

magnetic pressure

experiments on eachtwo or three

two or four panels;

27

Laser flyer

transparentsubstrate

carbon (0.5)

aluminum (0.5)alumina (0.5)

aluminum (4.0)

coatings (thickness in microns)

laserpulse

flyer(up to ~1mm thick)

accelerationby expandingplasma

Design: D.L. Paisley, P-24

28

Laser flyer

spacer ring

(dried coffee)

working fluid

(molasses)

10 mm

substrate

(soda-lime glass)flyer

(copper)

assembly

(view through substrate)

29

Radiative ablationMaterial ablation drives shock wave by reaction and plasma pressure.

referencematerial

sample

radiatio

n d

rive

sho

ck transit tim

em

easurem

ent

Radiation source: nuclear explosion, laser hohlraum, laser ablation.

30

Other shock diagnostics

surface velocimetry, radiography, transient x-ray diffraction

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X-ray streak camera

X-ray streak camera

laser beam

to drive shock

crystal

laser beam

to generate X-rays

point source

of X-rays

Bragg reflection(s)

Laue reflection(s)

31

Surface velocimetry

Doppler shift of reflected laser light.

Example velocity history as shock reaches surface:

velo

city

time

elastic precursor

plastic shock

spall and ringing

32

Transient x-ray diffraction

Si crystal, (100) orientation, 40 µm thick by 10 mm across:

33

Quasi-isentropic compression

drive beamilluminates targetto generate dynamic load

plasma blowofffrom surfacesupports compression

loading wave

sample

compressed state

window(optional)

velocimetry

34

Quasi-isentropic compression

Si, TRIDENT shot 15018:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14

irrad

ianc

e (P

W/m

2 ), v

eloc

ity (

km/s

)

time (ns)

laser irradiance

free surface velocity

30 µm 59 µm

35

Shock Hugoniot data

Acetone:

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

shoc

k sp

eed

(km

/s)

particle speed (km/s)

TruninMarsh

36

Empirical mechanical EOSFunctional fit to us – up data, e.g. Steinberg polynomial:

us = c0 +

n∑

i=1

si

(

up

us

)i−1

up,

R-H relations for states on the Hugoniot. Additional assumption to calculatestates off-Hugoniot: Gruneisen approximation,

p(ρ, e) = pref(ρ) + Γ(ρ)[

e − eref(ρ)]

.

Thus

p(ρ, e) =F (ρ, e)

H2(ρ, e)+ [Γ0 + bµ(ρ)] ρ0e

where µ(ρ) = ρ/ρ0 − 1,

F (ρ, e) = ρ0c20µ {1 + µ [(1 − Γ0/2) − µb/2]}

H(ρ, e) = 1 + µ − µ

[

n∑

i=1

si

(

µ

µ + 1

)i−1]

(thermodynamically incomplete).

37

Shock Hugoniot fit

Acetone:

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

shoc

k sp

eed

(km

/s)

particle speed (km/s)

experimental datacubic Grueneisen

38

Accuracy of empirical EOS

Shock Hugoniot for solid and porous Al; empirical EOS fitted

to us – up; Γ(ρ) estimated from slope of us – up:

0

2

4

6

8

10

12

0 1 2 3 4 5

shoc

k sp

eed

(km

/s)

particle speed (km/s)

2.785 g/cm3

2.50 g/cm3

2.25 g/cm3

2.00 g/cm3

1.70 g/cm3

39

Accuracy of empirical EOS

Shock Hugoniot for solid and porous Al; empirical EOS fitted

to us – up; Γ(ρ) adjusted to reproduce porous Hugoniot points:

0

2

4

6

8

10

12

0 1 2 3 4 5

shoc

k sp

eed

(km

/s)

particle speed (km/s)

2.785 g/cm3

2.50 g/cm3

2.25 g/cm3

2.00 g/cm3

1.70 g/cm3

40

Accuracy of theoretical EOS

Shock Hugoniot for solid and porous Al; theoretical EOS uses

ab initio quantum mechanics:

0

2

4

6

8

10

12

0 1 2 3 4 5

shoc

k sp

eed

(km

/s)

particle speed (km/s)

2.785 g/cm3

2.50 g/cm3

2.25 g/cm3

2.00 g/cm3

1.70 g/cm3

41

Limitations and current research

• Thermodynamic equilibrium

– electronic states: ‘non-LTE effects’ (below ns)

– atomic configuration: phase change dynamics (ps to

Myr)

• Stress tensor: decomposition into EOS and stress deviatornot always valid.

• Measurement of temperatures in shock experiments is

difficult – how to test theoretical EOS?

• Accuracy of quantum mechanical EOS predictions,especially for f-electron materials (lanthanides and

actinides).

42

Stress in continuum mechanics

strain (including compression)

block of matter

heat

What’s the stress? (tensor)

Strain tensor e(~r), stress tensor τ(ρ, T, e) – generalized EOS.

Deviatoric decomposition:

e = µI + ε, τ = −pI + σ; µ ≡ 1

3Tr e, p ≡ −1

3Tr τ

Assumption that scalar properties are independent: p(µ, T ),

σ(µ, T, ε).

43

Uniqueness of the EOS

QM prediction of stress tensor in Be for different elastic strains:

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

mea

n pr

essu

re (

eV/A

3 )

mass density (amu/A3)

a=2.29 Aa=2.20 A

D.C. Swift and G.J. Ackland, Appl. Phys. Lett. 86, 6 (2003).

– at the end of the day, the EOS may not really exist!

44