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Optical radiation from shock-compressed materials and interfaces
OPTICAL RADIATION FROM SHOCK-COMPRESSED MATERIALS AND INTERFACES
Bob SVENDSEN Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA Jay D. BASS Department of Geology, University of Illinois, Urbana, IL 61801, USA and Thomas J. AHRENS Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
I
NORTH-HOLLAND AMSTERDAM -
PHYSICS REPORTS (Review Section of Physics Letters) 180, No. 6 (1989) 333—416. North-Holland, Amsterdam
OPTICAL RADIATION FROM SHOCK-COMPRESSED MATERIALS AND INTERFACES Bob svENDsENa)*, Jay D. BASSb) and Thomas J. AHRENsa) Seismological Laboratory, California institute of Technology, Pasadena, CA 91125, USA ~ Department of Geology, University of Illinois, Urbana, JL 61801, USA Received April 1989
Contents: 1. 2. 3. 4.
Introduction Model considerations Model assumptions Initial conditions for conduction and radiation 4.1. Shock-compressed state 4.2. Release and reshock states 4.3. Application to Fe targets 5. Conductive transport in the target 6. Radiative transport in the target 7. Models and data
335 335 338 339 340 350 354 357 365 367
8. Summary Appendices A. Shock front and target boundary conditions B. Equilibrium thermodynamics and shock compression C. Isentropic release and reshock D. Energy transport in the target E. Conductive transport model F. Radiative transport model References
375 377 377 387 391 395 401 407 414
Abstract: Recent observations of shock-induced radiation from oxides, silicates and metals of geophysical interest constrain the shock-compressed temperature of these materials. In these experiments, a projectile impacts a target consisting of a metal driver plate, a metal film or foil layer, and a transparent window. We investigate the relationship between the temperature inferred from the observed radiation, and the temperatures of different high-pressure states (e.g. shocked state) of the shock-compressed film (or foil) and window. Deviations of the temperature in each target component away from that of their respective shock-compressed states occur because of (1) shock-impedance mismatch between target components, (2) thermal mismatch between target components, (3) surface roughness at target interfaces, and (4) conduction within and between target components. In particular, conduction may affect the temperature of the film/foil material at the film/foil—window interface, a major thermal radiation source in the target, on the time scale of the experiments. To be observed, radiation from sources at the film/foil—window interface or in the shocked window material must propagate through (1) the shocked window material, (2) the shock front, (3) the unshocked window material, and (4) the unshocked window free surface. Consequently, the observed intensity of target radiation sources is affected by the optical properties of each region. In particular, the source radiation intensity may be greatly reduced due to absorption in the shocked and/or unshocked window material, and/or only partial transmission through the film/foil— window interface, shock front and/or unshocked window free surface. To illustrate various aspects of the model, we apply it to radiation data from targets composed of an Fe driver plate, and Fe film or foil layer, and either an Al 203 or LiF window layer. * Present address: Department of Mechanics, Darmstadt Institute of Technology, D-6100 Darmstadt, FRG.
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B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
335
1. Introduction Traditional studies of the behavior of shock-compressed materials assess the mechanical response of these materials to shock compression (e.g. the change of density with pressure). Since this approach cannot directly constrain the temperature of the high-pressure state, other means are needed to provide experimental constraints on a complete equilibrium thermodynamic description (i.e. pressure—density-temperature) of these materials. To this end, recent experimental efforts observe and record shockinduced radiation; both initially transparent materials (e.g. alkali halides, summarized by Kormer [1], A1203: Urtiew [2], Si02 and Mg2SiO4: Lyzenga and Ahrens [3]) and (2) opaque materials at interfaces viewed through transparent or semitransparent windows (e.g. Mg: Urtiew and Grover [4], Ag: Lyzenga [5], Fe: Bass et al. [6]) have been investigated. This radiation, at least at high pressure, appears to be dominantly thermal, and so constrains some temperature in the target. In this paper we explore possible relationships between the experimentally constrained temperature and the temperatures of different high-pressure states achieved in the target components and at their interfaces during the experiment. We attempt this in the context of a simple model of energy transfer and transport in the targets, first dealing with conductive transport, and then radiative transport, which may be considered separately for the experimental conditions and materials of interest. We begin by discussing how shock compression, release, reshock and phase transitions may influence the temperature profile in the target (section 2), arguing in particular that these processes operate on a much shorter time scale than energy transport by conduction and/or radiation (section 3). We then discuss the likely importance of radiation versus conduction (section 3, appendix D) to the target energy balance via dimensional analysis, which implies that conduction will likely dominate radiation as a means of energy transport at high pressure in the materials of interest here. After these general considerations, we begin the detailed development of (1) the initial conditions for energy transport in the target as established by shock compression, release, possible reshock and phase transitions (section 4, appendices A, B, and C), and separate (2) conduction (section 5, appendix E) and (3) radiation (section 6, appendix F) models based on idealizations of the target geometry. Finally, we discuss the possible use of the combined conduction—radiation model, and certain simplifications of it, to fit data (section 7).
2. Model considerations Consider the target depicted in fig. 2.1, representative of that used in the experiments of Lyzenga [5] and Bass et a!. [6]. This generic target consists of (1) a 1.5mm thick metallic “driver” plate (DP), (2) a metallic film (1 to 10 ~imthick) or foil (10 to 100 ~imthick) layer (FL), and (3) a dielectric, transparent window (TW, 3 to 4mm thick). The target is constructed such that the shock impedance (i.e. the product of the starting density and shock wave velocity) of the DP is greater than or equal to that of the FL, which in turn has a shock impedance greater than or equal to that of the TW. An edge mask (fig. 2.1) prevents the detectors from sensing radiation from the edge of the target assembly, where conditions are probably inconsistent with the model for shock compression used in this work (see appendix A). Radiation emitted from the center of the assembly, where uniaxial compression most likely occurs, reflects from the mirror into the detectors. The experiment begins when a projectile impacts the DP (fig. 2.1), generating a shock wave that propagates through the DP to the DP—FL interface. Since this interface is formed by mechanical
336
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Pyrometer
Driver PI~1~
~Mirror
~
Driver Plate
Film Driver Material
-
Window
Material
Interface
Foil or Film
Fig. 2.1. Target assembly.
juxtaposition of the metallic DP and FL surfaces, it is “rough” on a —1 ~m scale (Urtiew and Grover [7]). The shock front thickness is ~0.01 p~min the materials and at the pressures of interest (e.g. Kormer [1]). With respect to the shock wave, then, the DP and FL surfaces are, prior to compression, partially free. Consequently, the shock wave accelerates the DP material at the DP—FL interface (which compresses the DP—FL interface) and simultaneously reflects from the DP surface at the DP—FL interface as a release wave propagating back into the DP, which releases the DP to near-zero pressure. The moving DP surface impacts the FL surface, generating shock waves of approximately equal magnitude that propagate backward into the just released DP, and forward into the unshocked FL. The former shock wave compresses the just released DP material from its low-pressure, high-temperature release state to one with approximately the same pressure as its previous compressed state; wave reverberations quickly bring this DP state to a state with a pressure equal to that of the compressed FL and a temperature well above that of the previous (i.e. first) DP compressed state. If the backward propagating shock wave overtakes the release wave at some distance behind the DP—FL interface, this distance defines the thickness of a reshocked DP material layer at the DP—FL interface. However, if the release wave is faster than the reshock wave, the entire DP may experience low-pressure release and reshock. In either case, subsequent wave reverberations quickly bring the DP to a state with the same normal (to the interface) material velocity and stress fields as the shocked FL. Since the DP material accelerating across the DP—FL interface impacts a rough FL surface, a thin (on the scale of the surface roughness) layer of film or foil material compresses, much like a porous material (Urtiew and Grover [7]), to a much higher temperature than achieved by the compressed FL material beyond this zone. As with the DP material at the DP—FL interface, the shock front traversing the FL reflects from a partially free surface at the FL—TW interface as a low-pressure release wave and accelerates the FL material across the interface to impact with the TW material. Since the TW surface at this interface is smooth relative to the shock front thickness, and the TW is much, more incompressible than either the DP or FL, the impacting FL material should not heat a thin layer of TW material, but rather only shock the TW up to high pressure and its Hugoniot temperature. The encounter of the shock wave propagating through the FL with the FL—TW interface generates
B. SventLsen et al., Optical radiation from shock-compressed materials and interfaces
337
backward and forward traveling shock waves, and the former wave compresses the low-pressure, high-temperature, released FL material to a state with approximately the same pressure as the first FL compressed state; wave reverberations quickly bring this FL state to a state with a pressure equal to that of the compressed transparent window (shocked window: SW) and a temperature much higher than the first FL compressed state. If the backward propagating shock wave overtakes the release wave, it cuts off the zone of release / reshock in the FL material. In this case, the combined wave releases the remaining FL material, and then the DP, to a state with approximately the same normal velocity and stress as the SW. Alternatively, if the shock wave does not overtake the low-pressure release wave, the entire FL and/or DP is released and reshocked. In either case, subsequent wave reverberations should quickly bring both the DP and FL to states possessing normal stress and material velocity fields equal to those of the SW. As the FL material at the FL—TW interface is compressed, released, and possibly reshocked, it heats up and begins to radiate. Consequently, the observed radiation intensity rises sharply (fig. 2.2A,C). As the shock wave travels forward into the TW, the thickness of the SW increases (fig. 2.2B); consequently, so does its contribution to the total observed radiation (note increase with time in fig. 2.2C). If the SW is highly absorbing and/or scattering (apparent in many experiments, Boslough [8]), the radiation intensity from the interface will decay with time (fig. 2.2C, dash—dot curve labeled fast decay); if not, the interface source will dominate the observed radiation history (fig. 2.2C, continuous curve labeled slow decay) since the FL material at the FL—SW interface is almost certainly at a higher temperature —Shocked Film — Partially-Released —
i—
Driver ‘Plate
~
1
(A)
Release Front
/
SW
FL
/~
x
“‘e
I NT ~
TOTAL foot decoy
~
~
decoy
I dec
0
~
I
slow decoy
Enters Target” ~
~
+
~sIow
Shock Wave //
~
/iSW
Shock Front
rs
I
DP
I
‘INT
Shock:d —Window ~a~riol
~
~:~j-~
Film
Film-Shocked Window Material Interface
i~~fl
~
Shock Wave Reaches Interface t — t INT
(C)
TW (B)
=
Fig. 2.2. Dynamic model geometry. Shock front reaches film/foil layer (FL)—transparent window (TW) interface (x 0) at time t1, when radiation is first detected. Interface radiation (lINT) dominates the early radiation history (A). If (1) the interface temperature decays slowly, and (2) the FL—TW interface, shocked window and shock front remain relatively transparent, LINT will dominate the observed radiation history during the experiment (“~5~ + slow decay” curve). However, if the FL—1’W interface and/or shock front develop significant reflectivity, and/or the SW develops significant opacity, ~INT will decay quickly (dash—dot curve), and may fall below the radiation intensity of the SW, ~ on the time scale of the experiment. The total intensity is then represented by the “i~~ + fast decay” curve.
338
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
than the SW. Assuming the observed radiation represents the sum of the interface and SW contributions, the interface contribution will dominate the observed radiation intensity unless the SW is strongly absorptive. Since the reshocked layers at each interface are significantly hotter than the surrounding material (see Urtiew and Grover [7], and discussion below), the temperature and radiation histories of targets with smooth versus rough interfaces should be quite different, and perhaps even experimentally resolvable. We investigate the dependence of the radiation history on the nature of the interface by examining radiation data from both mechanically formed foil—TW interfaces and vacuum-coated film—TW interfaces. In particular, we expect the vacuum-coated film—TW interface to be ,much smoother than the mechanically formed foil—TW interface. This expectation turns out to be somewhat naive, however, as shown below. Since the TW surface at the FL—TW interface is smooth (as defined above), we presume that any roughness of this interface is due to roughness of the FL surface. Besides conduction, dynamic phase transitions or other energy sources and/or sinks present in the FL, FL at the interface, and/or SW on the time scale of the experiment may also introduce time dependence into the temperature and effective emissivity inferred from the radiation observations (Grover and Urtiew [9]). In this study, we limit ourselves to the processes of conductive and radiative transport. With respect to the latter process, we must account for the effects of propagation through the SW, shock front, unshocked window (USW), and the TW free surface on the FL—SW interface and SW source radiation intensities.
3. Model assumptions We assume all sources contributing to the observed radiation intensity are thermal and in local thermodynamic equilibrium. We can then relate the source intensity to the wavelength A, and absolute temperature T, through the Planck function IApl(A, T), given by I5~1(A,T) where C
5(eC2~T—
1)’
(3.1)
A
7Wm2/sr and C 1 ~5.9544 X i0~” 2 1.4388 X 10’~mK are constants. Comparison of the observed radiation wavelength dependence with that of a black-body source, as represented by 15P1(A, T), implies that materials shock compressed to ~70 GPa are dominantly thermal radiators (Lyzenga et al. [11], Boslough [10]). At lower pressures, however, most materials apparently radiate both thermally and nonthermally (Si02: Kondo and Ahrens [12], Brannon et al. [13], Schmitt et al. [14]). Note that several of these materials are initially dielectric (e.g. Si02). The microscopic processes responsible for the radiation from these materials (defect electronic transitions?) are presently unidentified, but may be implied by spectrometric observations (Kondo and Ahrens [12]). In principle, energy transport in the target occurs by both radiation and conduction; our task is much more difficult if both radiation and conduction contribute equally to this process. In simple terms, we can estimate the likely relative contribution of radiation and conduction to energy transport within layers and across interfaces via dimensional analysis. The relevant nondimensional number is known as the Stark number, Skr (Siegel and Howell [15]), referenced to some state r, and given by (eq. D.15)
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Sk
=
r
339
krlXTrIXr arxrnrusaTr
This number represents the ratio of conductive to radiative flux, whether across a layer or within an “infinite” medium. It is composed of the materal properties kr~a~and n~,i.e. the thermal conductivity, radiation absorption coefficient, and refractive index, respectively. The remaining parameters, which may be material, include x~,Tr and z~ T1, the governing length scale, a reference temperature and 2 K4 is the Stefan—Boltzmann constant. temperature range, respectively. ~ we5.6696 10~W/m For the layered geometry of theAlso, target, may xassociate Xr with the layer thickness, Tr with the compressed or released temperature of the layer, and ~ Tr with the change in temperature across a given layer in the target such that L~Tr/Xrreflects the magnitude of the average temperature gradient across the layer. Via Skr~ we see that radiative transport dominates conductive transport in (1) an optically thick (a~—*oo), (2) a poorly conducting (k~—*0)and/or (3) a high-temperature medium, all other parameters being finite. Applying this parameter to the balance of energy in a target consisting initially of a metallic DP and FL (e.g. Fe), and a dielectric TW (e.g. A1 3 for the SW and the we find (appendix that 5k,. D). 10 and —i0 (compressed/released/reshocked) DP or FL,203), respectively In addition, viewing each layer as an infinite medium implies that conduction affects the balance of energy in both the FL and SW over a length scale of =
—
4. Initial conditions for conduction and radiation Within the assumption that (shock) compression, release, and reshock affect the temperature field on a much shorter time scale than conduction, these former processes establish the initial temperature
340
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
distribution in the target. Consequently, we begin constructing the model by estimating the temperature changes in each target component due to compression, isentropic release, phase transitions and/or reshock.
4.1. Shock-compressed state
For a given target material, let
{T1, P1, v1, p~,s1, e~} represent its uncompressed, or initial thermodynamic state, as characterized respectively by the temperature, pressure, material velocity, mass density, specific entropy, and specific internal energy of the material in that state. Likewise, let 5H’ eH} represent its compressed, or Hugoniot, state. For most (if not all) shock compression {TH, P11,experiments, VH, PH’ we have v~= 0 m/s, T 1 = 298 K and P1 = 0.1 MPa. With the constitutive assumptions that each target material (1) compresses adiabatically and (2) compresses as an ideal fluid, we obtain, in part A.1 of appendix A, the balances of mass (eq. A.2), linear momentum (eq. A.22), and energy (eq. A.30) across the shock front, u
=
—
~[‘(v11
—
v1)
.~
(4.1)
,
(4.2) —
e1
=
(2pI)~(PH+ P1),
(4.3)
respectively, where ij 1 PI’PH is the relative compression, (u v~)is the shock front propagation velocity (with respect to the uncompressed state), and u is the shock front displacement (i.e. “absolute”) velocity. With the further constitutive assumption that (u v1) is a linear function of —
—
—
(VH
—
(4.4)
u—v—a1+b1(v11—v1)
(e.g. Rice et al. [20]), and the initial state known a priori, (4.1)—(4.4) represent 4 equations in 5 unknowns: PH’ u, VH, P~’and ef{. In (4.4), a~and b1 are, respectively, the intercept and slope of the U~VH relation referenced to p1 and v~.Substituting (4.4) into (4.1), first for (u v~)and then for (vH v1), and then substituting the resulting two expressions into (4.2), the balance of linear momentum across the shock front takes the form (McQueen et al. [21]) —
—
2. (4.5) P1 = p1a~i~/(1b1ij) Relation (4.5) is known as the shock wave equation of state. The validity of (4.5) is clearly limited by the validity of the constitutive assumption (4.4), and the fact that (~H F 1) as b1 ~ 1. U—Va data for Fe (Brown and McQueen [22]) and many other materials (see Marsh [23]) appear to be well fit by a linear relation between u and VH. Relations (4.1)—(4.4), and so any relation directly derived from them, such as (4.5), are independent of any thermodynamic considerations, such as thermodynamic equilibrium and phase transitions. In particular, if compression induces a phase transition of some phase a, stable at T~and P°,to some other phase 1~,stable at T~and P~,we may write (4.1)—(4.5) in the forms —
—
—
—~
—+
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
341
~
(4.6)
P~ P~° = p~°(u v~)(v~ ~:°), —
—
—
(4.7)
—
t°(s~’, p~’)=(2p~)1’q~(P~+ Pr),
(4.8)
e~ase~(s~, p~)—e
~
(4.9) a
a
a
a2
2
‘~H’~1=pj(aj)~~/(1—bji~~) , (4.10) respectively, where ~ as 1 p°/p Although shock compression itself is a non-equilibrium, irrevers—
~.
ible process, it is still possible that (4.6)—(4.10) connect states of thermomechanical equilibrium, i.e., that the initial and Hugoniot states represent states of uniform, constant temperature, pressure, chemical potential and material velocity. In this context, we may connect the initial and Hugoniot states as distinct equilibrium states via a classical thermodynamic path (i.e. a path connecting a series of equilibrium states) representing a change in specific internal energy equal to that determined by (4.8), the balance of energy across the shock front, as follows (McQueen et al. [21], Ahrens et a!. [24]). Define z~eP0aseP(sp,p~) e’°(s~°, p~0)aseP e° —
(4.11)
—
as the difference in specific internal energy between a at T~and P~,and 13 at T~and
~p.
Since all of the high-pressure properties and fields appearing in the relations to follow in this section belong to 13, we drop the 13 superscript on these for simplicity in what follows. If we compress 13 isentropically from its initial density pf to its Hugoniot density PH’ the change in its specific internal energy is
~eSHase(sP,pH)_e(sP,pP)__f~
P 5(p)dlnp,
(4.12)
where P5( p) as P(s~,p) is the pressure as a function of density along the 13 isentrope referenced to s~. Since this last state and the Hugoniot state are at the same density PH’ we may connect them by an equilibrium thermodynamic path at constant density or volume. With de = T ds = c~d T at constant density or volume, where c~,is the specific heat at constant volume of 13, define
~eVH e(sH,
PH)
—
e(s~,PH)
=
J
cvH(T) dT
(4.13)
T011
as the difference in specific internal energy between 13 states at the same density PH~In (4.13), c~/54is the specific heat of 1~at a density PH’ TSH is the temperature of the 13 state at a density PH along the 13 isentrope referenced to s~,and TH is the temperature of the 13 state with specific internal energy equal to (4.8). TSH may be obtained from the relation (c9lnTIdlnp)5=y,
(4.14)
342
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
when given a knowledge of y, the equilibrium thermodynamic Grüneisen parameter of (4.8) and (4.11)—(4.13), we obtain
TOH
cvH(T) dT = —(~e~~ +~
+
fla(~H
Combining
(4.15)
Pr),
+
13.
a relation for the equilibrium thermodynamic temperature of a state of 13 with specific internal energy equal to that of its Hugoniot state. If we assume that y of 13 is a function of density alone (e.g. Boehler and Ramakrishnan [25], Anderson [26]), then the relation py(ôe/ôP)~= 1 for 13 at PH yields =
PHYH with
~
as
TSH=
(4.16)
(~H — ‘~sH)’
PS(PH) and YH
as Y(PH).
In this case, we also have from (4.14)
TPexp(fy(P)dlnp).
(4.17)
Combining (4.8), (4.11), (4.12) and (4.16), we obtain [1 (1 + 2YH)fla](PH —
—
‘)~)= (1
—
71co)(1~sH—
‘)~) p~yH(~eI+ ~e5~), —
(4.18)
which is similar to the form used by Jeanloz and Ahrens [27]. Both (4.15) and (4.18) represent the balance of energy across the shock front, assuming that the compression process connects states of thermodynamic equilibrium. However, since (4.18) depends upon the assumption that y is a function only of density, while (4.15) does not, these two relations are not completely equivalent. We use (4.15) to calculate T11 as a function of or PH, once we have relations for ~ zXe511, cr11 and TSH. Recall that, to obtain (4.18), we compressed the material from a state to ~H’ incurring a phase transition in the process. Imagine now that we can compress 1~from a state ~ to ~ For this case, (4.1)—(4.5) become 1(vH—vP), (4.19) ~H
.~
u—vP=n~ —
P~= p~(u v~)(v —
11 v~), —
—
e~= (2pP)~~(PH + Pr),
(4.20) (4.21)
(4.22) 2m~/(1 br~)2, (4.23) pr(ar) respectively, with as 1 PP/PH. Relations (4.19)—(4.23) represent a generalization of the “meta—
—
=
—
B.
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
343
stable” Hugoniot concept of McQueen et a!. [21] to an arbitrary initial state Z~.We emphasize that = Z~is by definition the same state in both (4.6)—(4.10) and (4.19)—(4.23). Combining (4.8), (4.16), and (4.21), we obtain [1—(1+ 2yH)fl13](PHPI)—(lTh3)(PSHPI)pIyH~eSH,
(4.24)
which is analogous to (4.18), and may be obtained from (4.18) when we replace p°with p~,P~with P~, and set ~ = 0. Noting that
2(ôe/e~p)f= p2(d z~e~/dp),
(4.25)
P(sf, p) p (4.24) may be rearranged to give as
PH(~)
—
YH AeSH = ~j
[1 (1+ 2YH)~l13](PH —
P~)+
—~--
(4.26)
P~,
i.e. an ordinary differential equation for a a a l3 13 z~esH—1Xesu(pH;pi,a1 ,b1,p~,‘y1),
when P~is given by (4.10). We solve (4.26) numerically, subject to the initial (1) ~eSH(pP)= 0 [by definition, i.e. (4.16)], and (2)
(PH
=
p~)conditions that
p~(dL~eSH/dPH)~ ~ = from (4.25). Note also that consistency between (4.18), as based on (4.6)—(4.10), and (4.24), as based on (4.19)—(4.23), requires that as
P5H(PP)
2= P =
PH(pP)
=
P~+ (ib~)
1
+
2p~~
(4.27)
with ~ as 1 p~0/p~~ Clearly, assuming p~ p~° and ~ 0, we have P~ P~°. The change in specific internal energy along the 13 isentrope referenced to s~,AeSH, is usually calculated via some “equation of state” P(s~,p) for 13. In most cases (e.g. finite strain: Stacey et al. [28]), the resulting relation for ~eSHtakes the form —
=
~eSH(pH,
p~,K~,K~’),
where K~and K~’are the isentropic bulk modulus of 13 and its first pressure derivative, respectively, referenced to Z~.In fig. 4.la we compare values of i~e0~ for r-Fe calculated via (1) eq. (4.26), (2) third-order spatial finite strain (e.g. Stacey Ct a!. [28]), (3) third-order Ullman—Pan’kov finite strain (Uliman and Pan’kov [29]), and (4) Murnaghan’s equation of state (e.g. Stacey et al. [28]). The parameters used for this calculation are listed in table 4.1. As noted by Somerville and Ahrens [30], the third-order Ullman—Pan’kov relation is compatible with the linear U~VHrelation (4.4), since both make similar predictions for higher-order derivatives of K~.Again, note that (4.26) and methods (2)—(4) give
344
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
0
90
a
180 270 Pressure (GPa)
360
0 0~
(S —
—
—
Third-order-FS UlImoo-Poo’kov
‘ ‘.‘ .
Energy-balance
‘ ,‘ —
,,,
~rnoghon1~
0
90
180 Pressure (GPa)
270
360
Fig. 4.1. (a) Comparison of different estimates of the change in specific internal energy (SIE) of c-Fe as a function of pressure along an isentrope referenced to ~ ~. These calculations are based on the parameter set for c-Fe given in table 4.1. “Energy balance” refers to eq. (4.26) in the text. (b) Comparison of different estimates for pressure along the isentrope, ~aH’ of c-Fe referenced to Z ~, based on the parameters for c-Fe given in table 4.1.
us
z~eSH= ~eSH(pH;
~
a~°, b~,p~,‘yr) and
~e
5~
=
~eSH(PH; p~,K~,K~), respectively. For con-
sistency and a true comparison, we constrain the values of K~and K~for e-Fe via the relations 2, (4.28) K~= p~(a~) when we assume that a~is equal to the bulk elastic wave velocity, and (Ruoff [40],part A.4 of appendix A) K~’=4b~—1,
(4.29)
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
345
Table 4.1 Standard temperature and pressure~~ (5TP) parameters Quantity 3) p (kg/rn a (m/s)
c-Fe 8352’~
liquid Fe 7952’°
Al
4487’~ 1.57’~ 168c)
4038’~ 1.58’~
8908° 0.91° 254~~
203
LiF
39~C)
2650d)
K (GPa)
5.28) cj,, (J/kgK) 1) ya (X10’ K q t,~(K)
130g)
5.31’~
432°~
~5i) 43k)
7750) 1.6~
1.95m)
L32m)
i.o’~
i.ø°~
1250’° 2345~
385D
TM (K)
18090)
k (WImK) 80~ i~(nfl m) 50°~ *) The reference pressure for a~and b~, and so for
46p)
0
K,
and
5050d)
~g)
4.28~ 10.3~
1615d)
1.78w)
580d)
845°~ 3p)
0
K~,of c-Fe and liquid Fe is actually P~,as
given by (4.27). 5T)(Drotning [32]). a) Jephcoat et al. 1311. b) Robie o) Calculated et al.from [33]. p(T) = 8136(1 — 7.608 x 10~ d) Van Thiel [34]. Estimated from u 3955 + l.S8vH (Brown and McQueen [22]).
=
Fit to Al
°
203 data in ref. [23]. ~ Calculated assuming K, = ~ 5) Anderson et al. [35]. ‘~Calculatedwith K~ = 4b, — 1 (Ruoff [40]). ~ Andrews [42]. ~Assumed the same as a-Fe (Touloukian et al. [36]). °
Touloukian et al. [36].
o)
Weast [37], p. D-187.
w) Calculated from y = aK,/pc,.. n) yp = const. assumed in all calculations. ‘~ Assumed
the same as a-Fe in Touloukian et al. [38].
q) Inferred from Keeler [39]. assuming only that 13 compresses as an ideal fluid. To use (4.28)—(4.29) for this purpose, we require values for a~ and b~.Again, since both (4.10) and (4.23) give us ~H = P~,via the definitions of a~and b~as coefficients in the metastable U~~VHrelation (4.22), we may use (4.10) and its density derivatives to estimate a~and b~,as shown in appendix B. From this procedure we obtain a~and b~ as functions of p~°, a~,b~and p~in the limit PH~~~PP. Leaving the details to appendix B, we have a~ (1 1 —
4
—
3]u2a~ , ~)[(1 + b~’~)/(1 b~~) 4(1— ~ (1- b~)(1+ b~)
(4 30)
—
which are (A.64) and (A.65), respectively. Note that, from these relations, af—* a°and b~—~ b~as p~—~p’, i.e., as t~—~0. With p~known a priori, and a~° and b~constrained from experimental U~~~VH data, then, we need only p~to obtain a~and b~from this method. The calculations presented in fig. 4. la demonstrate that ~e 5~predicted by each of these relations
B. Suendsen et al., Optical radiation from shock-compressed materials and interfaces
346
(except that of Murnaghan) is essentially the same. Note that the Murnaghan isentrope is off scale in fig. 4.la. Once we have ~ via solution of (4.26), (4.24) provides the “equation of state”, i.e., ‘~sH =
P~+ PHYH z~eSH+
[1—(1+ 2YH)’7’31(~H Pr). —
(4.32)
For the derivatives of ~eSH, of course, the minor differences between the different expressions for z~eSH are magnified, as shown in fig. 4.lb, where we plot ~sH as given by (4.32), and compare it with the values for ~sH given by the equivalent finite-strain and Murnaghan relations. In particular, note that the energy balance equation of state (4.32) is a bit “stiffer” than either of the finite-strain equations of state.
Since both dielectric and metallic solids initially compose our target, we must consider thermodynamic properties that reflect the influence of both ionic and electronic processes. To estimate the harmonic lattice contribution to these properties, we use the Debye model (e.g. Alt’shuler et a!. [41], Andrews [42]). Andrews [42] used this model as part of a parameterization of the equilibrium thermodynamic properties of a- and E-Fe. Jamieson et al. [43], Brown and McQueen [44, 22], and Boness et a!. [45]have all assessed the influence of electronic processes on the material properties of metals at high pressure and temperature. They assume the conduction electrons contribute to the equilibrium thermodynamic properties of a metal as a Sommerfeld free-electron gas (e.g. Wallace [46], section 24). This is reasonable for for T much than thethan Fermi Since this for 5 K, assuming that T~ Fe is less much less thetemperature. Fermi temperature of temperature Fe seems quite Fe is =i0 reasonable for the calculations presented below. Assuming that lattice-electron and band-structure contributions are negligible, the molar Helmholtz free energy (HFE), F(T, p), of a cubic or isotropic solid material subject to an isotropic state of stress is then given by (Wallace [46], section 5) F(T,p)=cP(p)+3PR[~D+ln(1—e~°)—~ED(~D)]T— ~1T2(p)T2.
(4.33)
Here 1(p) represents the zero-temperature lattice contribution to the molar HFE, =
~D(T, p)~eD(P)/T
(4.34)
is the Debye parameter, (l(p) as T(p)
—
2A 2(p),
(4.35)
2is the low-temperature (Tmuch less than the Fermi temperature) electronic contribution to F, ~F(p)T A 2 is the high-temperature anharmonic contribution to F, v is the number of atoms in the 2C°)formula, T chemical and ED(~)is the Debye internal-energy function (e.g. Gopal [47]), —
ED(~)~4JxXl
dx.
(4.36)
In the Debye approximation, the Debye temperature °D is related to a lattice Grüneisen parameter, YD’ via (Wallace [46])
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
347
(4.37)
YD =(dln OD/dln p).
If we assume
(4.38)
, YD(P) = YD(P~)(PIIPYt°
then OD is given by @D(Pi)exP(q
~D(P)
[1_~/D]).
(4.39)
For simplicity, we do not separate YD into longitudinal and transverse components; in this case, YD represents a weighted average of these components. The quantity E(p) is related to the electronic Grüneisen parameter, Ye’ via (Wallace [46]) Yeas —(din F/dln p).
(4.40)
Assuming Ye is constant, we have F(p)asF(p~)(p (4.41)
1/p)~. By analogy with (4.41), we assume for simplicity that
(4.42)
M. (1(p) as
(l(p1)(p1/p)
For the example calculations involving Fe targets presented below, we constrain the values of (1( p~)and w empirically. Boness et al. [45] calculated F( p) for the 6- and -y-phases of iron using the Sommerfeld free-electron gas theory. In addition, these authors suggest that the electronic density of states in liquid iron at high pressure may be approximated by that of the close packed 6- and y-phases at high pressure, where the liquid should be “close-packed”. We make the same assumption. Relation (4.33) allows us to write expressions for the approximate density and temperature dependence of a number of solid-state properties (appendix B). In particular, the equilibrium thermodynamic Grüneisen parameter, y, based on (4.33), is given by (eq. B.10) (4.43)
c~
where M is the molecular weight and cv the specific heat at constant volume. From this, we note that y is only weakly temperature dependent, since w YD in the pressure range of interest. Hence, the thermodynamic model based on (4.33) is approximately consistent with the assumption above that ‘ is a function of density alone, upon which (4.18) above is based. Consequently, we assume ~v YD(P)’ where YD(P) is given by (4.38). In this case, from (4.14) and (4.38), we also have T(s 1, p2)
=
T(s1, p1) exp( YD(Pi)
[(~/~)~D
—
(p./p2)~D])
(4.44)
348
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
for the change in temperature in the material due to isentropic compression or expansion from a starting density p1 to a second density p2 along the isentrope of the material referenced to s1. With respect to T~,we are particularly interested in the specific heat at constant volume, c~(T,p), given by c~(T,p)~—~(-~~)=~~ [4E1~(4~)—
~~]+
~
T.
(4.45)
Substituting this into (4.15), we obtain [E~(~1.~)T~ED(4D3)T$H] —
+
~-~,j(2H(T~
—
(4.46)
T~H)= ~eVFI,
12(PH). Note that i~eVH= 0 when TH = T with (l~ 311 and that TH> T311 when ~ >0. In (4.46), we use ~Ds ~DH’ TSH, ~DH ODH/TH and °DHas OD(PH). Equation (4.46) is an implicit relation for TH, which we evaluate numerically. Since the majority of our calculations are at high temperature (T> ~D), and the Hugoniot temperature changes much more drastically with pressure than ~ we 4DH—~O)form, may approximate ED(~DH) by its high-temperature ( fDH
ED(~DH~0)=(~3) 3 ~ ~
(4.47)
Substituting this into (4.46) and rearranging, we obtain
(4.48)
~
0DH + ~
~VH
+ ~
(4.49)
u2HT~H.
ED(~DS)TSH+ 9vR
Relation (4.48) has the solution TH=2v’~cos(~ cos1(q/pV~))—2vR/(2~,
2vR2 2M p=—(-~-—)+~-n—nH’ H
H
~
yR
2M
H
H
(4.50) 2
2vR2 H
If we set (l~equal to zero in (4.46), we have ED(~DH)TH((2H= 0) = ED(~DS)TSH+ ~
~
(4.51)
which is appropriate for a dielectric material with negligible anharmonic contributions to F. Doing this in (4.48), we obtain
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
M 1 3~R~9 21/2 TH(flH=0)=~—~A41+[1-~ MA~)]
(
}~
349
(4.52)
fll~as~evH + 3yR ~j_ ED(~D)TSH + 9~R ~ ~DH~ Relation (4.52) is appropriate for a dielectric material at high (T>
9D)
t
temperature with A
2T~H< A2T~‘~ 1. Alternatively, if we assume the classic limit for the harmonic contribution to cv at high temperature, i.e. 3vR/M, eq. (4.46) reduces to 3vR TH(~DH=0)=—fl --—
M12 2A~) LY~2~ (~R) [1’
\h/2
A~asi~ev~+ ED(~DS)TSH+ ~
—1], 1
(4.53)
QHT~H.
_~_
Finally, if we set both f2~and ~DH equal to zero in (4.46), we have TH(QH
=
0,
4DH =
0)
=
ED( ~Ds) TSH
+
3R z~e
(4.54)
5,54.
If we further assume ED(~D$) 1 in (4.53), we obtain the relation most commonly used (e.g. Jeanloz and Ahrens [27]) to calculate TH. To demonstrate the effect of these different approximations to Cv on TH, we plot TH as a function of pressure for Fe compressed from a-Fe to s-Fe in fig. 4.2, using the parameter set for Fe given in table 4.1. Below, we constrain fl~from the intersection of the Fe Hugoniot and melting curves, but for the purposes of this calculation we assume that A2 = 0 and use the results of Boness et al. [45] for F(p) (table 4.1). From fig. 4.2, we see that (l~is the dominant influence on the Hugoniot temperature of 2-Fe at high pressure, and that the difference between TH calculated with the full Debye relation for cv, 0 C,
0 C
Do
Debye-plus-electrons Debye-minas-electrons
0
a
DnIong-Pet~t-p1us-eIectrnns
-
co Du1ong-Petnt-~nus-elec1rnns
-~
~
0 0~ 0
Q)~t
—
___
a—
-
0
1
Melt,ng —
—
0
in
.—
1o
-
~1l~i
E~
0
60
120
180
240
Pressure (GPa) Fig. 4.2. Comparison of the effect of different models for the specific heat at constant volume, c~(T,p) on the temperature of the c-Fe Hugoniot states.
350
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
eq. (4.46), and TH calculated by assuming the harmonic part of Cv = 3vR/M, eq. (4.53), is very small (about 200 K at 240 GPa). In fact, even for Al2 03, which possesses a much higher Debye temperature than 6-Fe (table 4.1), Hugoniot temperatures calculated with eq. (4.51) are only about 300 K above those calculated with eq. (4.54) at 200 GPa. Note that the curves in fig. 4.2 converge at low pressure because T~approaches TSH “faster” than the various approximations to c~can affect TH as P~—~ P~. To summarize, from the impedance match and u—vu relations of each material, we may obtain the pressure and density of the first shocked state of each target component. Using this along with an estimate of ~ePh1 and the assumed form for ‘ above allows us to estimate, using eqs. (4.26) and (4.44), the changes in specific internal energy and temperature along the appropriate isentrope of the high-pressure phase of each target component. These estimates, along with the model (4.45) for c~(T,p), allow us to calculate TH for any phase as a function of P~or p11. The next step is to estimate the effect of release on the compressed state (TH, P~and PH). 4.2. Release and reshock states As discussed above, the targets are constructed so that the DP has a higher shock impedance than the FL, which in turn has a higher impedance than the TW. In this case, both the DP and FL are compressed, released and possibly reshocked. Assuming that a given release state of the DP or FL is in thermodynamic equilibrium, we may again employ the concept of an equivalent equilibrium thermodynamic path to connect respective compressed and released states of each target component. However, since we have no expression for the change in specific internal energy of the material during release that is independent of the details of phase transition [recall that eq. (4.3) is such a relation for shock compression], we cannot utilize the same kind of equilibrium thermodynamic path as that constructed above for shock compression. Instead, we must assume something about the release process, and any potential phase change during release, to construct an equilibrium thermodynamic path between the compressed and released states. The only constraint we have a priori is that the release process takes the compressed material at an interface, via the release path, to a state with approximately the same normal components of material velocity and stress as the compressed state of the lower impedance material on the other side of the interface. Subsequent wave reverberations establish the continuity of normal stress and material velocity, as required for the existence of a material interface. To proceed further, we assume that heat transport in or out of the target is insignificant on the time scale of the release process, i.e., this process is adiabatic. Considering each compressed target component as an equilibrium thermodynamic system, we further assume that any mechanical work by the system during release is entirely reversible. In the case of a single-phase system, the release path is then both isentropic and adiabatic. The change in temperature with density along this path is related to y, as given by (4.14) above. Since the impedance match provides us with the pressure of the release state, ~R, we may calculate the temperature, TR, and density, PR’ along an isentropic path that has not encountered a phase boundary, through simultaneous solution of (4.14) and
p(TR,
~R)
=
p(THR,
~R)
exp(_
f
a(T, p(T, ER)) dT),
(4.55)
THR
where TR
T(sH,
PR)
is the release state temperature,
PR as p(TR, PR)
the release state density, and
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
351
p( THR, PR) is the density along the (possibly metastable) Hugoniot of the same phase at a temperature THR and the pressure of the release state, ~ The coefficient of thermal expansion a( T, p) in (4.55) is derived from an equilibrium thermodynamic model for F( T, p) of the appropriate phase (appendix B for solid state, Svendsen et al. [48] for liquid state), since by definition (Wallace [46]) 2F/ôT9p)PT
2
aas 2p(9Fh9p)~+p p(ô (92F/ôp )T
(4.56)
as,
KT
where KT = KT(T, p) is the isothermal bulk modulus; for the solid-state model, it is referenced to the isentrope or Hugoniot, as discussed in appendix B. To bound the nature of the release process as initiated at an interface, we focus on the extremes: (1) complete contact (roughness ~ shock front thickness) at the interface, or (2) no contact, in which case each material has a free surface at the interface. We refer to the former interface as the “smooth” interface, and to the latter as the “rough” interface. To illustrate the different paths these “endmember” interfaces may take, consider the two examples discussed below and depicted in fig. 4.3. If we compress the DP (respectively, FL) to some point A along its Hugoniot below the Hugoniot—melting curve intersection, it will release to a state having, after one or two wave reverberations, the normal stress and material velocity of the compressed FL (respectively, TW). If these reverberations are isentropic, the resulting temperature will equal that calculated by direct release to the pressure of the compressed FL (TW). The compressed DP (FL) material at the smooth interface then releases directly to this state, represented by point B in fig. 4.3. However, the surface of the DP (FL) at the rough interface is partially free; hence, the compressed DP (FL) material at this interface releases to near-zero pressure. If we assume the release path is isentropic, its slope will be less steep than that of the melting curve. In this case, the melting curve and release path will intersect (point C, fig. 4.3). If the phase transition is slow relative to the rate of decompression, the releasing DP (FL) material will follow the (metastable) path ABCG to low pressure (even though this path is not necessarily isentropic, as discussed below). However, if the transition is uninhibited, the release path will turn along the phase
LIQUID
SOLID
Pressure Fig. 4.3. Possible range of temperature—pressure paths taken by DP and FL materials near the DP—FL and FL—SW interfaces, respectively, during an experiment.
352
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
boundary at the intersection point (point C, fig. 4.3), and the mixed-phase material will decompress along the boundary until the transition is complete, or the mixture reaches the release state. Assuming the transition completes above zero pressure at joint D in fig. 4.3, the now liquid DP (FL) leaves the phase boundary and continues to decompress along DE to P1. As the released DP (FL) material closes the interface, it impacts the FL (TW) material and is reshocked and reverberated along a series of paths, collectively symbolized in fig. 4.3 as the paths lying between EF and GH, up to the smooth-interface, release-state pressure, which is that of the shocked FL (TW). Note that the temperature achieved by this set of shock paths is bounded above by the temperature estimated from a single compression back up to the Hugoniot pressure of the FL (SW); we use this bound, along with isentropic release, to estimate reshock temperatures in what follows. Since the temperature of the released DP (FL) is much higher than the initial temperature of the unshocked DP (FL), the temperature of the reshocked DP (FL) material is higher than either the first shock state or the release state of the DP (FL) material at a smooth interface (point F versus point B, z~TRs,fig. 4.3). If the DP (FL) material is shocked to a higher pressure state than A but still below the melting curve, a smooth interface may release to a state pinned to the melting curve, or be above the melting curve, as for release from A’ in fig. 4.3. The rough interface released from A’ would follow A’B’C’ and be reshocked along C’D’ to D’. Note that the effect of reshock is much more pronounced as the pressure of the first shocked state increases, regardless of the phase transition. When the release path encounters a phase boundary, such as the melting boundary shown in fig. 4.3, eq. (4.14) is no longer valid. If we believe the release path remains isentropic through this region, then we must require that, in addition to releasing adiabatically and doing or experiencing only reversible work, the material also change phase in thermodynamic equilibrium (see appendix C). Under these conditions, the isentropic two-phase path for a congruent phase transition from phase 13 to phase ‘~ is described by the relation [(c~ + x ~c~) ~v
13v’° + x ~av)T~
—
8 ~s] dP + iXs TPB ~ d,~’= 0,
(a
(4.57)
where P is the pressure along the phase boundary TPB(P), x is the mass fraction of ii, c,,, is the specific heat at constant pressure, a isanythequantity coefficient of thermal expansion, v is Since the specific and is the jump of ~(‘ across the phase transition. (1) all volume, end-member quantities in (4.57) may be viewed as functions of pressure and temperature, and (2) temperature and pressure are not independent along the equilibrium phase boundary, these quantities are actually a function only of pressure or temperature along the boundary. In this case, choosing P as independent, we may solve (4.57) for x = ~(P) to obtain (eq. C.29) 13
—
~i
~(P)= ~(P)X(P)(~(PH)x(PH)x(PH)+
I
~(P*)(T
av ~s
—
c’3p ~v) dP*),
(4.58)
PB
P
liii
p~(P)~exptJ
—
Lz~av
—
~v/ —
TPB
(~1+
~c\1 ) —~--~)j dP*
1,
(4.59)
and X(P) ~s TPB(P) ~s. In (4.58), x(PH) is the mass fraction of rr (e.g. liquid) present in the Hugoniot state. We evaluate (4.58) numerically in the mixed-phase region along the phase boundary TPB(P) until (1) ~(P) + x(P11) = 1 (complete transformation) or (2) P = ~R (partial transformation). In
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
353
the former case, the new phase then releases to ~R along a path beginning at the pressure and temperature on the phase boundary where ~(P) + X(PH) = 1. Relation (4.58) is valid along any isentropic path through a first-order mixed-phase region of a single-component system (i.e. solid—solid, liquid—solid), but now we focus on the solid—liquid phase boundary as discussed above. To utilize (4.58), we need relations for the solid and liquid state properties along the solid—liquid boundary TM(P). We do this by way of semi-empirical models for the solid state (e.g. Andrews [42], and appendix B), and the liquid state (e.g. Stevenson [49], Svendsen et al. [48]), respectively. For the solid state, we use a parameterization of the solidus, TM(p~),based on Lindemann’s law, -
(dTM/dP~)Lindemann =
(4.60)
T~A~/p~,
where A~as2(ysM_~)
(4.61)
for the solid—mixed phase boundary (solidus). The quantity ~ is the solid phonon Grüneisen parameter at the melting point, equal to YD(P~)in the Debye approximation we use here. Using (4.61) in (4.60), we may calculate TM(p~)once we know the density of the solid along the phase boundary, and the dependence of A~on the solidus density. For the solid phase, we have already assumed as YD(P)’ with YD given by (4.38) above. Putting (4.38) into (4.61), and the resulting combination into (4.60), we obtain an expression for the solidus, i.e., TM(p~)as TM(p~M)(~~)exp{~~2~r [(-~)h1r
(4.62)
—
where p~11is the density of the solid at the melting temperature at standard pressure. To use (4.62) to find TM(p~),we need to calculate the change in density along the solidus with pressure. Noting that, since in general the equilibrium thermodynamic properties as developed from (4.33) are functions of temperature and density, we may calculate the variation of any of these properties i~r(T, p) with temperature at constant pressure from the relation ~(T, P~)= ~(Tr, Pr) +J(~)dT,
(4.63)
(d~/iIaT)~ = (9~~IaT)~ap(ôçfJIdp)~,
(4.64)
—
and ‘~rand Tr are some pressure and temperature at which we know cu. In particular, putting = p into (4.62), as we did to obtain (4.55) above, we may anchor the solidus TM(p~)to the solid Hugoniot of the relevant solid phase by solving (4.63) (numerically) simultaneously with ~i
p~(P)=
P(TH,
P) exp(_f a(T, p(T, P)) dT),
(4.65)
B. Svendsen
354
et al., Optical radiation from shock-compressed materials and interfaces
C
C
_____
C
C C ~
C
C -
CC -
=
C C
- - -
C. C
“
.
-
-
- -
•1-
—
--
H -=
C C C —
C ___________
0
90
—
oolidus-compression Hugoniol-compression
_________________
180 Pressure (GP)
270
360
Fig. 4.4. Comparison of c-Fe Lindemann solidi calculated from the compression along the Hugoniot (dotted curve) with that estimated from the compression along the Hugoniot adjusted to the solidus temperature (dashed curve), as discussed in the text.
where p(T~,P) is the Hugoniot density of the solid phase at the same pressure. In fig. 4.4, we compare this calculation with one in which we assume P(TM, P) = p(TH, P). The greatest effect is at low pressure; this is also where the correction is most uncertain. For the rough DP—FL and FL—TW interfaces, the DP and FL release completely, and consequently we cannot use the Hugoniot as a reference state. So instead of (4.55), we solve (4.14) simultaneously with P(TR,
0)
=
P(Tr,
0)[1
—
a(Tr, 0)(TR
—
Tr)],
(4.66)
where Tr is some reference temperature (e.g. 298 K or TM), depending on the relevant phase. With the density of the release state, PR’ we may estimate the free surface velocity of the DP and FL surfaces at the DP—FL and FL—TW interfaces, respectively, due to isentropic release, via the Riemann integral method (Rice et al. [20]). We assume, as required by the constraint of isentropic release, that the material velocity is continuous across the phase boundary (i.e. the same for both phases) when calculating the free surface velocity. We then take this free surface velocity as the “projectile” velocity of the DP (FL) surface impacting the FL (TW) surface, and use an impedance match to calculate the pressure and density of the reshocked state. To calculate the temperature of the reshocked state, we use the appropriate form of (4.46), but referenced to the temperature and density of the complete release state rather than to T,~and p~. 4.3. Application to Fe targets To exemplify these considerations, we calculate release and reshock states for Fe—Fe—A1203 and Fe—Fe—LiF targets, as shown in figs. 4.5a,b. The solid and liquid T~,P11 states result from (4.46) and (4.53) with A2 = 0 and ~Ds = 0, respectively, as based on the parameter set given in table 4.1. Solid-state properties along the release path and melting curve are referenced to the r-Hugoniot via
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces C C 0 I
C .x,
~
—
Lindemana-melting
aa
Hugarnot-states Release-states
00
o
::
::~::i:::eton
~
Foil-date
0
-
000
~
o
0
I
Relesse-eonduct,on
-
355
Liquid
00 000
80
160
240
320
Pressure (GPa) 0 0 CJ __
I —
— 11)
an
Hugoniot-atates
no
Reshoek-states
ax
Reshoek-conductian
~
Fail-data
0
ci
I
Lindemwnn-melting
Otis 00
Liquid
xi’,
0
0
x~
I~0t0tt1huit0annaaad
0
80
160
240
320
Pressure (GPa) Fig. 4.5. (a) Release/reshock calculations for Fe film/foil—Al 203 interfaces and initial greybody temperatures inferred from Fe—Fe film/foil—Al203 radiation data of Bass et at. [6]. “Release conduction” and “reshock conduction” symbols represent initial effect of thennal inertia mismatch across the Fe film/foil—Al203 interface on the indicated states. (b) Release/reshock calculations for Fe film/foil—LiF interfaces and initial greybody temperatures from Fe—Fe flim/foil—LiF radiation data. The larger shock impedance mismatch between Fe and LiF results in a lower release-state pressure at Fe—LiF interfaces than at Fe—Al203 interfaces, when both release from the same Hugoniot pressure.
(4.55), (4.62) and (4.65), while the analogous liquid-state properties are referenced to the highpressure Fe melting data of Williams and Jeanloz [50] (as reported in Williams et al. [51]) via a liquid-state model for Fe (Svendsen et al. [48]). In calculating release paths, we ignore all other solid phases of Fe, save r-Fe, which is the stable solid phase of Fe along its Hugoniot between 13 GPa (Barker and Hollenbach [52]) and ~—~200 GPa, where the sound speed measurements of Brown and McQueen [44,22] along the Fe Hugoniot suggest that 2-Fe transforms to y-Fe (?) or possibly a new solid phase 0 (Boehler [53]). Consequently, -y-Fe and / or another solid phase is in equilibrium with
356
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
liquid Fe above about 5 GPa to perhaps 280 GPa (e.g. Anderson [26]). In this case, we neglect any effects of an r -y or r 0 transition in referencing compression along the Fe melting curve to the 6-Fe Hugoniot. As stated above, in calculating the 6-Fe Hugoniot states shown in figs. 4.5a,b, we have constrained fl(p~), with w = 1.34, which is the value of ~ for 6-Fe given by Boness et al. [45], by requiring the parameterized Fe melting curve and 6 Hugoniot to intersect at 245 GPa. On the basis of the parameter set given in table 4.1, this fit constrains fl(pf) to be 0.046 i/kg K2. Boness et al. [45] calculated a value of 0.090 i/kg K2 for F(p~)(adjusted to STP density for 6-Fe given in table 4.1). If we set fl(p~) = 0.090, the 6 Hugoniot based on the parameter set in table 4i intersects the melting curve at =280 GPa. We note that Boness et al. [45] constrained F(p~)= 0.09 J/kg K2 and Ye = 1.27 for -y-Fe, while Bukowinski [54] constrained F(p~)=0.08J/kgK2 and Ye = 1.5 for this phase. With these values for F(p~),the value of fl(pf) constrained above for 6-Fe implies some competition between anharmonic and electronic contributions to the specific heat of 6-Fe at high pressure. Brown and McQueen [22] fit a linear u—vH relation to the available Fe Hugoniot data between 13 and 400 GPa. Since their sound speed measurements also suggest that Fe melts along the Hugoniot above about 245 GPa, their u—vH relation should describe the liquid—solid mixture and pure liquid phase, as well as the solid. On this basis, we use their u—v 11 relation to calculate both the 6-Fe Hugoniot and a metastable liquid-Fe Hugoniot referenced to the extrapolated density of liquid Fe at STP (table 4.1). With this u—v11 relation, eq. (4.53) above for TH (A2 = 0 and ~Ds = 0), and F as constrained by Boness et a!. [45], we calculate the metastable Hugoniot of liquid Fe. Using ~ = 0.14 Mi/kg for Fe (as compared to the enthalpy of melting at standard pressure, 0.25 MJ/kg, from Desai [55]), the metastable liquid-Fe Hugoniot intersects the melting curve at about 305 GPa. This agrees reasonably well with the results of Young and Grover [56], who also ignored all other phases of Fe save 2 and liquid in their parametrization of the Fe melting curve. We combine this metastable Hugoniot along with the 6-Fe Hugoniot in an ideal mix (e.g. Watt and Ahrens [57]) to construct the compressed, mixed-phase region shown in figs. 4.5a,b. For comparison with the calculations, we have plotted the initial (t = 0) interface temperatures inferred from the Fe film/foil radiation data of Bass et al. [6] in figs. 4.5a,b. Note that the Fe—A1203 interface data shown in fig. 4.5a run almost parallel to the reshock locus, thereby exemplifying the strong pressure dependence of the reshock process (Urtiew and Grover [7]). Comparing the data with the smooth-interface release states, shown as squares in fig. 4.5a, implies that Fe at both film—A1203 interfaces experiences up to —1800 K of reshock heating between 190 and 230 GPa. As stated above, we naively expected that the film—TW interface would experience consistently less reshocking than the foil—TW interface. The present results contradict this expectation. There appears to be no guarantee that film interfaces will consistently experience any less reshock than the foil interfaces, especially at high pressure. In this case, a well-polished foil surface may actually experience less reshock than a slightly porous film interface. Figure 4.5b displays the results of the calculation for Fe—LiF interfaces. Because of the larger impedance mismatch between Fe and LiF, the Fe—LiF interface reaches a lower release-state pressure than the Fe—Al203 interface, given that both release from the same Hugoniot pressure. The data and calculation imply that lower release-state pressure results in less extreme reshocking. Note that the Fe—LiF and low-pressure Fe—A12O3 data fall right on the corresponding smooth-release locus. The points labeled “release conduction” and “reshock conduction” refer to the effect of the contrast or mismatch in “thermal inertia” across the Fe—TW interface on the release and reshock temperatures, as discussed in the following section. —~
—~
B.
Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
357
5. Conductive transport in the target
We assume that the temperature profile created by shock compression, release and/or reshock is established on a time scale sufficiently short to represent the initial conditions for energy transport in the target. Urtiew and Grover [7] considered the problem of energy transfer at material interfaces and demonstrated that a rough (~1 lim) interface experiences a higher degree of shock heating than a smooth (~1 ~m) interface, much like a porous material experiences relative to its crystalline counterpart. Since the TW surface at the (FL—SW) interface is much less rough (~108m) than the FL surface at the interface, it should experience little, if any, direct reshock heating. However, the DP and FL surfaces at the DP—FL interface, as well as the FL surface at the FL—TW interface, may experience significant reshock heating, as discussed above. Following Grover and Urtiew [9], we assume that (1) energy transport is parallel to the direction of shock propagation (i.e. one-dimensional), (2) both temperature and heat flux are continuous across each interface in the target (appendix A.2), and (3) there are no sources or sinks of energy in any layer or at the interfaces between them. Under these conditions, we may solve the one-dimensional energy balance (e.g. Carslaw and Jaeger [58]) pc~dT/dt = k d2T/dx2
(5.1)
for the temperature profile T = T(x, t) in each target component as a function of position along the direction of shock propagation x and time t. The time t = 0 corresponds to coincidence of the shock front and the FL—TW interface. In eq. (5.1), p is the density, c 0,, is the specific heat at constant pressure, and k is the thermal conductivity. Since the temperature profile in the FL, and particularly the temporal variations of the temperature at the FL—SW interface, control the intensity of thermal radiation sources at the FL—SW interface, we emphasize these in what follows. We expect a layer of the DP material at the DP—FL interface and a layer of the FL material at the FL—SW interface to experience some degree of reshocking. Also, the rough FL surface at the DP—FL interface should compress into a thin layer with a much higher temperature than the compressed solid FL material. With this structure, the initial (t = 0) temperature profile in the shocked target is of the form (fig. 5.1) T(X,0)asTD, as TD+L~TDF,
—(d+ÔD)
asTF+E~TFD, —d
asTF+i~TFW, asT~,
—(d—ôFD)
(5.2)
ÔFW
0
Here, d is the FL thickness, while TD and TF are the temperatures achieved in the DP and FL, respectively, by direct release to the pressure of the SW, which temperature T~.Also, TD + ~while TDF 8D inhas thea DP at the DP—FL interface, isTFthe temperature of the reshocked layer with thickness + ~ TFW is the temperature of the reshocked layer with thickness 6FW in the FL at the FL—SW
358
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Driver
Film / Foil
I
I I 5) -d
I
I
~FW
~
I
I
-(d+8
Window
-(d-&~0)
x
Fig. 5.1. Initial conditions for thermal conduction in target. T0 and T,, represent temperatures achieved in the DP and FL, respectively, upon direct release to a state with the pressure of the SW, having a temperature T~.Variable degrees of reshocking are shown for DP (~T0~) and FL (~T~0) at the DP—FL interface, and for FL (~T~~) at the FL—SW interface; these involve some thickness (h0, b~0,and 6~~) of each target component adjacent to the interface.
interface. If the surface of the FL at the DP—FL interface is also rough on some scale, it will compress like a porous material. Consequently, we assume that a layer with a thickness ~FD and temperature TF + ~ TFD forms in the FL at the DP—FL interface. Since the surface of the TW is much smoother than the DP or FL surfaces, we assume that there is no reshock heating of the SW material at the FL—SW interface. Note that the DP and SW are idealized as thermal half-spaces, a consequence of our assumption about the rates of compression and release relative to conduction. Again, we emphasize that all material properties of each target component are assumed homogeneous and time independent, and are referenced to their respective states at the pressure of the FL—SW interface. The governing relation (5.1) for each layer, combined with the boundary conditions of continuity of heat flux and temperature, and the initial conditions (5.2), specify an initial-boundary value problem for T( ~, T) in the target, where ~ = x/d and T = t/texp~We are particularly interested in this profile for the FL layer (—1 ~ 0), and the temporal variation of T(0, T), which represents the FL—SW interface temperature. Solving this initial-boundary value problem in appendix E, we obtain [eqs. (E.22)—(E.25)] TF(~,T) = TF
+
A(~,T)I~TDF+ B(~,T)(TD
—
TF) + C(~,r)z~TFD
+D(~,r)~TFw+E(~,T)(TF—Tw),~
‘
m
V ~
UDF
—
A(~,r)= 11
~DF)
(PDF~WF)
m’O
x [erfc{[(2m + 1) + ~]w2} erfc{[(2m —
.~
~DF)
as
(
1)
+ ~ + KFD~]wF}]
+
1)
—
(~DF~WF)~WF
m0
x [erfc{[(2m + 1) B(~,T)
+
m
V
cTDF — ~
(5.3)
T>O,
—
flw2} erfc{[(2m —
m[erfc{[(2m
~ (~2vw5)
(TDF) naO UDF
+1)
+
~
~]wF}
+
KpD~]WF}]
—
~WF
erfc{[(2m + 1)
—
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
g~,r,—1, ~)+g~,r, 1~i + ÔFD),
~,—1+ôD),
~:1+~FD,
C(~,r) as g~, T,
D(~,r) as ~ ~
r,
ôFW,
0),
359
—1 +
(5.4)
~FD’
+
~~FW’
E(~,T)as ( +~WF~WF)
~ ,n=O
(~FPWF)m[~Ferfc{[2(m+ 1)+
~]wF}
—erfc{(2m
—
~)wF}].
The functions g~( ~, ‘r, a, b) are defined in appendix E (eq. E.26). In these expressions, KFD as is the square root of the ratio of the FL thermal diffusivity to the DP thermal diffusivity, with K as k/pc~, ~ as x/d is the nondimensional distance, d is the thickness of the FL, ~ 6 as \/PeF/4r, 2/KFtexp is the Péclet number of the foil/film layer, rtexp as tItexp is the nondimensional time scale, PeF as d is the time scale of the experiment, ~ as 6 2~/d, and 8D as ÔFD/d. Also, we have ~
2, °~DF as
~WF as (kWPWcPW/kFpFcPF)112,
KF/ KD
(5.5a,b)
(kDpDcpD/kFpFcpF)U
which are the thermal inertia “mismatches” (Carslaw and Jaeger [58],p. 321) between DP and FL, and SW and FL, at the pressure of the FL—SW interface. Also we have ~w 2—(crw2
l)I(u~2+ 1),
PDF_(UDF1)/(UDF+
1).
(5.6a,b)
In (5 .5a,b), k, p and c~,are the thermal conductivity, density and specific heat at constant pressure, respectively, of the designated material for the state of at the pressure of k, thep SW. To 0W6 at high pressure, weeach needmaterial the appropriate values of and c~,. estimate follows the values tTDF and Density fromofthe impedance match and release calculations, while the specific heat at constant pressure results from the classical thermodynamic models discussed above. Assuming that the thermal conductivity, k, may be written in terms of lattice, k~,and electronic, ke, components, i.e. k = k~+ ke, we assume that k ke for metallic target components. In this case, we calculate kD and kF as a function of temperature from the Wiedemann—Franz—Lorenz relation (e.g. Berman [59]) = 2.45 x 10-8 W fl/K2,
(5.7)
where Pc is the electrical resistivity, which we estimate from electrical resistance data on shocked metals, respectively. Assuming the thermal conductivity of the TW material is controlled by lattice processes, we may use the thermal conductivity model of Roufosse and Kiemens [60] to estimate k~. As compared to kD or kF, k~ predicted from this model increases much more slowly with pressure, partially accounting for the development of a significant thermal inertia mismatch across the FL—SW target interface. Based on the release/reshock calculations presented above, we calculate 0~WFfor the Fe—Al 203 and Fe—LiF interfaces using the value of electrical conductivity for 2-Fe given in table 4.1. This value comes from Keeler [39], who summarized electrical conductivity data on 2-Fe between 20 and 140 GPa, and it represents an extrapolation of the trend in the c-Fe data down to standard pressure. We list results of this calculation in table 5.1. As is evident, Fe is more closely matched to Al203 than LiF; since T1(r) is proportional to o~~I(l + o-~~) [see eq. (5.8) below], there is a greater adjustment of T1(r) at the Fe—A1203 interface, as is shown below.
B.
360
Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
lable 5.1 STP and high-pressure thermal-inertia mismatch estimates
STP
100 GPa
200 GPa
ideal interface Fe—Al
203 Fe—LiF
0.56 0.20
0.25 0.11
0.15 0.05
Fe—A1203 Fe—LiF
0.56 0.20
reshocked interface 0.15 0.07 0.08 0.01
We now focus on the FL—SW interface temperature T1(r) as TF(O, r) = T~(0,T), since it is responsible for controlling the interface source radiation intensity. Specializing the relation for T1(r) to the case where the DP and FL are the same material (e.g. Fe), we have TD = TF, °DF = 1, and so ~DF = 0 via eq. (5.6b); we also assume that iXTDF = ~TFD for simplicity. In this context, the same equilibrium thermodynamic state is assumed to exist on either side of the DP—FL interface, and T1(r) is given by T1(r)
G(r)
TF+G(r)ATDF+D(r)~TFw+ l+UWF (T~ TF),
=
1+1
{erfc[(1
—
~D)wF]
—
erfc[(1
+
ô~)w2]},
°WF
(5.8)
D(r) = 1+1 ~WF
erfc(8WwF).
(5.9)
Noting that erfc(x) increases, while erf(x) decreases, as x decreases, we see that D(r) will decay with time, while G(r) can either grow or decay. Most radiation observations constrain a decreasing temperature with time (see discussion below); however, there may be some suggestion of the influence of ATDF on T1(r) in the data on the Fe—LiF interfaces discussed below. If we assume z~TDF= 0 = z~TFDand/or 5~= 0 = eq. (5.8) reduces to T1(r) = TF
+
D(r) z~TFW+ 1
+UWF
(T~ TF), —
(5.10)
which is the reshock model considered by Grover and Urtiew [9]. Further, if we let 6~—+0, we have, from (5.10), TJ(T)-+T!dasTF+
(T~—TF),
(5.11)
relating the temperature of the smooth FL—SW interface, TId, to the temperature 0WF~~ of the ~direct-release Clearly T state, TF. Note that TId = T1(c’D) and that TId approaches TF as ~r~2—*0, and T~as 1, as given by (5.10), will be time dependent only if the FL—SW interface is reshocked. We use (5.10) with the reshock 0D2 stateand temperature in the FLTat the FL—SW interface (TF + ~ TFW), the temperature of the 0wF’ to calculate SW (T~), 1(0), which is labeled “reshock conduction” in figs. 0WF~ 4.5a,b. As Similarly, we use (5.11) to calculate the “release conduction” temperatures from TF, T~and stated above, the Fe—LiF thermal mismatch is greater (i.e., ~ is much smaller, see table 5.1) than that of the Fe—Al 203 interface, mainly because LiF is more compressible and less conductive
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
361
(thermally, see table 4.1) than Al203. In this case, the Fe—LiF interface temperature remains closer to the temperature of Fe at the interface than does the Fe—A1203 interface temperature. Further, the greater compressibility of LiF gives it a much higher compressed temperature than A1203. For example, TH for LiF [via eq. (4.51)] at 160 GPa is ~4200 K (ignoring the possibility of melting), while TH for A12O3 [also via eq. (4.51)] at 230 GPa is ~~c2750 K. The temperature mismatch is much less across the Fe—LiF interface, and the effect of thermal inertia mismatch on T1 is less extreme. In figs. 5.2, 5.3, 5.4 and 5.5, we present calculations for TF(~,T) and ~ r) from (5.3) and (E.27), respectively, and the associated T1(T) = TF(O, T), with ZXTDF = Z~TFD,etc., as assumed to write (5.8), for Fe—Fe—SW targets. To construct these figures, we calculate the compressed/released and reshocked/released states achieved in an Fe—Fe—A12O3 target impacted by a Ta projectile at a velocity _______________________________________________ gi Os Film/Foil Window 0
to300-ns
a -1 ND Distance C C ~ F-
J Pedal ,asam5a,~l.O R.shockad ley.rO,1
000a
Interlace-I t°O
DODD
tinlinity
—
—
C I_C
2 .-‘
I.
‘~-
-
to
~oaoooo*OaoooOOoooaOooooOGOoOaooaaooooOOeOOO0OOOOOO
0. at
c 000000000000D00000000000000000D00000D00000000000000
b I
~. 1
75
150 Time (ns)
225
300
Fig. 5.2. Variation of the temperature near a reshocked Fe film/foil—A1203 interface. (a) The variation of the temperature in the FL, ~ r), and SW, ~ r), as a function of nondimensional (ND) position, ~, with respect to the FL—SW interface (~ = x/d = 0) at four different times during a 300 ns experiment. The nondimensional range of—ito 1 corresponds to —d to d, where d is the thickness of the FL layer. “Reshocked layer” refers to the nondimensional thickness of the reshocked layer, 8~. (b) The corresponding variation of the interface temperature, T1(t).
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
362
Film/Foil
Window
C ~-
g
Initial-eanditians
—
-
‘,
~.
t3-ass
—
..
- -
t30-ns
--
t300-ns
.-..
ar
C 1
-i
0 ND
1
Distance
C 0
I
0,1 N
-
Pactel ,ssamRar=1 0 R5s5ocked leye,’=0.5
—
— ~
DODD
~.
lnterface-T tiflfixs~ty
C C
2°a.’ 10
-
0
~
c
C C
a—
0,
—
—
—
— —
-
00000000000000000000000D0000000000
DODD DO DODODO00000
b o
75
iSO Time (ns)
225
300
Fig. 5.3. Variation of the temperature near a reshocked Fe fihn/foil—Al,03 interface. The reshocked layer depicted in (a) is thicker than that of fig. 5.2, and relative to the conduction length scale (as represented by the Péclet number), causing the temperature of the reshocked layer, and so that of the FL—SW interface, to decay more slowly. In (b), the “t = 0” curve corresponds to T1(0), while that labeled “t = infinity” corresponds to T~(cc). The magnitude of these asymptotic values of T1(t) is governed by that of the FL—SW thermal-inertia mismatch, FrWF.
of 5.67 km / 5; we assume that the calculated reshock temperatures at the DP—FL and FL—SW interfaces are the initial values. This impact velocity is that of one of the experiments (Fe—Fe film—Al203) discussed below. The basic result here is the dependence of the rate of change of T1(r) on PeF and 6~. From (5.8), the change in T1(r) with time is given by dT1(r)/dr
G’(T)~TDF+D’(T)~TFw,
(~) p
G’(r)
1 2 ~WF
D’(r) =
l+tTWF —2
(5.12)
1/2
[(1
-
~FD)
exp{-[(l
-
7TT
4’~TT (PeF)
2]. u26~
exp[—(~wwF)
~D)wF]2}
-
(1
+
6~)exp{-[(l
+ 6*)]2}]
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces I
I
Film/Foil
Window
~~ ~ g
363
-
—
—
- -
Initial-canditians t3-ns
----S.-
.....
-
.....
t30-ns t300-ns
~
a
I 0
C~
-i
1
ND Distance C
I
I
I
N
aneu — —
Rechecked Pactel num.be,=O.1 Leejee=0.5
DODD
I_
t=O Intertece-T tlnfinity
C C 10 0000*ooooooo000a000aoa000000000000000aa000000a0000
a.--
O
DD000Dn0000n000000D000000D000
00000 00000000000000000
C C
0
75
I
150
Time (ns)
I
225
C
3bt’
Fig. 5.4. Variation of the temperature near a reshocked Fe film/foil—A1203 interface. In this figure and fig. 5.5, we hold the reshocked layer thickness constant and vary the Péclet number, or conduction length scale, ~ of the FL. Since the Péclet number is inversely proportional to the conduction length scale, a relatively small Péclet number (0.1) results in a fast decay of T1(t), as shown in (b).
For the particular case we have plotted, and as noted above, unless ~ L ~ 1, and/or i~TD~‘z~TFW,the ~TFw term dominates T1(r). Since the coefficient of the dominating term, D’(T), is always negative, the rate of change of T1(r) will be negative, and T1(r) will consequently decrease with 2),with time. the D’(r) term in (5.12) is dominant, dT1/dT is of proportional to texp~ exp(—~ as ~ Further,_when FWVPeF/4r = ôFw/2\/~. In other words, over the time scale the_experiment ~ represents the ratio of the layer thickness ~FW to the conductive length scale VKFtCXP. Note that ~t exp(— ~2) achieves a maximum value near /L 1 and is much smaller (—0) for 1u much greater or less than unity. In figs. 5.2 and 5.3, we hold PeF constant (i.e. the conductive length scale \/K2t05~) and vary the layer thickness 6~. As shown in fig. 5.2, with PeF = 1, T1(r) for a thin reshocked layer ~ = 0.1) relaxes very quickly (i.e. faster than can be resolved experimentally) to near T1(cx~),while in fig. 5.3, we see that a thicker reshocked layer (~~ = 0.5) will relax much more slowly, and on a resolvable time scale. A similar set of events holds if we fix the layer thickness and vary the conductive length scale of —~
——
364
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces I
-
Film/Foil
Window
--
i—
g
—
—
initial-conditians
- -
ta.3-n, t35-ns
--
t300-ns
.,
—
-
a
0--i______________________________
-l
0 ND Distance
C N Pactel rnsraber—10.0 Renhacked tasjer—0 5
_
0000
interface-I t=ø
DODD
ta.infmity
—
—
C 0
a I..
G-G.eG4G.~Q.Q000.50o00oo0oo0oo000oooooo00oo00ooooo00o
0.
—
-
-
-
-
-
a) Io
C
0 00000000000000000000
D000D 00000000 0000000 DODODO00000
0
0
76
150
225
300
Time (ns)
Fig. 5.5. Variation of the temperature near a reshocked Fe film/foil—Al,0
3 interface. With a Péclet number of 10, the conduction length scale of the FL is small relative to the thickness of the FL, and the interface temperature decays very little over the time scale of the experiment.
the FL, as we show in figs. 5.4 and 5.5. For a conductive length scale large compared to the reshocked-layer thickness (fig. 5.4, PeF = 0.1 and ~ = 0.5), T1(r) relaxes relatively quickly, whereas if the conductive length scale is small relative to the layer thickness (fig. 5.5, PeF = 10 and 6~ = 0.5), 5~P trade off there is little or no resolvable relaxation of T1(T) away from T1(0). Obviously, ~FW and V in their effects on T 1 (T), introducing some ambiguity; only their ratio has a distinct effect on T1 (T). In any event, for PeF 1 and intermediate ~ —0.3—0.7) reshock-layer thicknesses, T1(t) is time dependent on an experimentally resolvable time scale, and its variation with time produces a corresponding radiation source time dependence, as we show in the next section.
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
365
6. Radiative transport in the target With a model of the initial temperature profile of the target components and interfaces, we now establish a connection between the radiation intensity of sources at these temperatures and the radiation intensity emerging from the free surface of the TW during the experiment. The target is represented as a series of plane-parallel layers (fig. 2.2) with optically smooth boundaries (Boslough [8], and appendix F). We assume that (1) source radiation is collimated by the target geometry, (2) all radiation sources are thermal, and so their intensity is given by the Planck function, (3) sources are located only at the FL—SW interface and/or uniformly throughout the SW, particularly along the direction of shock propagation, and (4) all optical properties are independent of wavelength. The model spectral intensity of the radiation emerging from the free surface (FS) of the USW (unshocked window), ‘~mod= IAmod(1k’ t), as a function of wavelength, A, and time after the shock front has passed the FL—SW interface, t, is given by (eq. F.43) 1’Amod(’~, t) as êAsw(t)IAPl(A,
T~)+ êAl(t)IAPI(A, T 1(t))
.
(6.1)
The Hugoniot temperature of the SW, T~,is assumed homogeneous, uniform, and constant, which is consistent with a uniform distribution of radiation sources in the SW. The interface temperature, T1, is a function of time, or constant, in the context of the conduction model discussed above. The subscript A denotes a spectral quantity. In (6.1), we identify ~Asw(~)as ~t’~(t)[1 TASW(t)J[l
+
—
rAITASW(t)]
as ~l~(t)rAsw(t)(1— rAl)
(6.2) (6.3)
as the effectve normal spectral emissivities of the SW and FL—SW interface, respectively. As is evident from (6.1), ~ASW and ~AJ are the properties connecting the intensities of the sources within the target and the intensity emerging from the target. The function ~P,5(t)is defined by as
(1
—
rAFS)TAUSW(t)(1
—
TAsF)
,
(6.4)
and represents the effect on source radiation of propagation through the FS, USW, and SF (shock front). In eqs. (6.2)—(6.4), rAFS, rASF and r~1are the effective normal spectral reflectivities of the FS, SF, and FL—SW interface, respectively. Further, TAUSW(t) as
TASW(t) as
exp[—a~~~~(1 t/texp)1 , —
exp(—a~5~t/t0~~) ,
(6.5) (6.6)
are the effective normal spectral transmissivities of SW and USW layers, respectively. The quantities ~ and a~sware nondimensional forms of the effective normal spectral absorption coefficients in the USW and SW, respectively, and they are given by a~usw
_
aAUSWxFS,
(6.7)
B.
366 I
*
aASW
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
=
.‘
V~\yt~xp)
aASWtSXFS
respectively, where VHW is the material velocity of the SW, and ~ and aASW are the dimensional counterparts of ausw and ~ respectively. Also, XFS is the thickness of the TW, and so the position of the FS with respect to the FL—SW interface (fig. 2.2). Note that ~ and a~uswmediate the explicit time dependence of ~ and ~AI. In writing (6.7) and (6.8), we have also assumed steady shock propagation such that the position of the shock front in the TW (fig. 2.2) may be written xsF(t) as (u
—
v
11~)t(appendix F), where From eqs. (6.7) and (6.8), ~ aASW
(xFs
—
VHWtexp)
VHW
‘
is the material velocity of the SW. and ~ will be of order unity when
a~J~,Z
XFS
‘
(
.
)
which are both —i~~ m~,since the thickness of the TW is generally —i0~m. So for values of aASW and/or aAUSW much larger or smaller than these “geometric” values, source radiation intensity is resolvably or not, affected by propagation through the SW and/or USW, respectively. The USW is usually transparent, so aAUSW 0; if the SW is transparent as well, then aASW 0 and, from (6.2) and (6.3), we have ~AI(~)
~ EAI(O) =
(1
—
TAFs)(1
—
rASF)(l
—
TAI)
,
(6.10)
and ~Asw = 0, respectively. In this case, ‘Amod is governed entirely by sources at the FL—SW interface, and any time dependence of the observed radiation history is due solely to T1. Note that the bound on ~Al in (6.10) is also the initial value of ~AI (i.e., ~Al can only decrease with time). If the TW becomes opaque upon shock compression, we have aASW —+ Again, with aAUSW 0, we have EAI —30 and ~.
EAsw(t) <(1
—
TAFS)(1
—
rASF)
(6.11)
.
In this case, observable sources are confined to the shock front (this is the “ideal case” of Boslough [8]). The impact of these and other model parameters on IAmod(A, t) is more explicitly depicted by writing the partial derivatives of IAOsOd(A, t) with respect to A and t. From eqs. (6.1), (6.2), and (6.3), these are A(l9IAmOd/t3A)t = texp(ôIAmod/19t)A
T~)+ P(~)êAI(t)IApl(A,T1(t))
~ IAmoda~Usw+ {[1
—
+ P(pII)êAI(t)IApl(A,
r~5+ 2rAITASW(t)]IAPI(A, T~)
—
T1(t)) dIn T1/dt
,
—
SIAmod(A,
êAI(t)IApl(A,
t)
,
(6.12)
T1(t))}a~~
(6.13)
P(~)as~t/(1—e~), ,t~~asC2IAT~, ~asC2/AT1. Relation (6.12) exemplifies the fact that the wavelength dependence of ‘Amod is due solely to that of the Planck function, since we have assumed that the optical properties are independent of wavelength. We make this assumption because it is not clear at this point that existing data can resolve wavelength dependent optical properties (Svendsen and Ahrens [61]). Again, for most TWs, we have aAUSW —0. If, in addition, T1(t) is approximately constant with time,
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
367
which may occur in a thick (PeF ~‘ 1) or thin (PeF ~ 1) FL with a thick (6~ 1) or thin (5~ °~1) reshocked layer of FL material, or at a smooth interface, as discussed above, (6.13) reduces to ——
texp(t9lAmod/8t)A
=
{[1
—
2rAI’rASW(t)]IAPI(A,
TAI +
T~)
—
EAI(t)IApl(A,
T 1(t))}a~~~ .
This will be positive if 2rAITASW(t)]IAPI(A, T~)>êAI(t)IAPI(A, T [1 r~1+ 1(t)) ,
(6.14)
(6.15)
—
but otherwise negative, since a~swis always Consequently, finite andofa TAmodpositive. will grow or decay withwith timea on the value basis of of ~the sign time independent interface temperature, ‘ (6.14). If the TW is initially transparent and remains relatively transparent upon compression, we have ~ and ~ Putting these into (6.13), we have texp(ôIAmod/t9t)A
=
P(p~J)ê~ 1I~~1(A, T1(t)) d ln T1/dt
,
(6.16)
and any variation of ~ with time should reflect that of T1 through the Planck function. In particular, ‘Amod will increase or decrease as T1 increases or decreases at a given wavelength. 7. Models and data We compare models and data in the context of the standard al. [63]). In our case, it is given by N~
N 1
as
2 statistic (e.g. Bevington [62],Press et
x
1 —~
i-i j=1 ~
2. [~A~X~(A~, ~)
—
IACOd(A~,t1
a)]
(7.1)
In this relation, 1~exp(Ai,ti), 1Amod(Ai, t 1 a), ~ as o-(A1, t1) are, respectively, the experimental and model spectral radiances, and the experimental uncertainties, all at a particular wavelength, A1, and time, t1. Also, NA and N~are the number of wavelengths and times sampled, respectively, in the experiment. The five-component “vector” a is the model parameter vector, in our case given by
a as {rASF, ~
T~,r~1,T1(t)}
.
(7.2)
Since the radiation model is nonlinear with respect to ~ a~sw, T~and T1(t), (7.1) is not strictly a maximum likelihood measure, even if the data errors are normally distributed. However, if the best-fit values of the a (i.e. amjn) have uncertainties sufficiently small such that the value of ‘Amod can2(a) be well will approximated thethe first two termslikelihood of its Taylor series(Press representation about IAmod(A, t; amin), ~ be “very close”byto maximum estimate et al. [63]). We note that TAFS and ~ are not included in (7.1), since they may be calculated or determined from index of refraction and absorption data for the TW. From the conduction model, we have explicit expressions [e.g. eq. (5.8)] for the time dependence of T 1, which allow us, in principle, to constrain 0~WF’etc., given fitted values of T 1. Similarly, the fitted reflectivities allow us to constrain changes in the indices of refraction across boundaries (e.g. the shock front). Note that, in general, the optical
368
B. Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
properties constrained from (7.1) will not be A-dependent unless we give them, a priori, an explicit A-dependence, with constants whose values are chosen by the fit (i.e. by the data). Since we have no reasonable expectation for this A-dependence, we cannot truly constrain it. It is for this reason, plus the limited resolving power of the data itself (Boslough [10], Svendsen and Ahrens [61]), that we assume a~~\\T, aAUSW, etc., are independent of A in the previous section. However, we may determine an apparent A-dependence of the optical properties if we specialize (7.1) and fit at each wavelength over time, i.e.,
I
I 000a) 450-nm DO~ 600-nm t2~) 750-nm .aSA~900-nm
to l.a
o%
E
0
0
a
0
-
r
II
0
0
260
130
390
Time (ns) c_i Wien —
b
GS
— GSel --LM
40
200
120
20
Time (ns) Fig. 7.1. (a) Observed radiation history and (b)
x2
statistic of greybody model (ê
15, Tg0) applied to radiation data from Fe—Fe film—Al203 target impacted by a tantalum projectile at 5.67 km/s, resulting in an Fe Hugoniot pressure of 244 GPa and an A1203 Hugoniot pressure of 190 GPa, which 2. (c, Greybodyatmodel is also the Fe—A1203 interface pressure (Bass et al. 161). Part (a) displays the radiation intensity (spectral radiance) data,d) collected four wavelengths: 450, 600, 750 nm. Part “goodness” the greybody fit, emissivity, as indicatedé~by Chi2 = x parameters constrained by and data 900 displayed in (b) (a); displays (c) the the best-fit normal of greybody effective 0(t);(d) the corresponding greybody temperature, T10(t). Fits using Wien’s law, Golden Search (GS), GS with the effective emissivity set to 1 (GSeI), and the Levenburg—Marquardt algorithm (LM) are indicated.
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
369
1 ~ [IACXP(AI, t x2(A~a) as j=l ff~ 1)
2 —
jAmod(Ai, t1
(7.3)
.
a)]
Svendsen and Ahrens [61] constrained akmjfl in this manner for radiation data from Ta—Ag—MgO targets. We use a very simple version of this approach below with the data of Bass et al. [6] to constrain aASW.
First, however, it is instructive to consider fits to data using simpler models than that represented by
(7.1) and (7.3). Most earlier workers (e.g. Kormer [1], Urtiew [2], Lyzenga [5], Lyzenga et al. [11]) constrained model parameters via the greybody relation iAgb(A, Egb~ Tgb) as EgbJAPl(A, Tgb)
The associated
x2
(7.4)
.
statistic is given by
Wien —
C
GS
--LM
40
120
200
28u
Time (ns) c
cj
I
I Wien
~ 40
1~0 200 Time (ns) Fig. 7.1. (Cont.)
d
2~u
370
B.
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
1
NA
Xgb(tj; Egb~Tgb) =
i=l
(7.5)
t 2. [1~exp(Ai, 1) JAgb(Ai, tj; Egb~ Tgb)]
~
—
Since the summation in (7.5) averages ~g 5over all observed wavelengths, it represents a wavelengthaveraged effective (i.e. total) emissivity. Given that the only A-dependence in the greybody model is contained in the Planck function, IAPI(A, T), the more closely the data follow the blackbody wavelength distribution at a given temperature, the better the fit [i.e. the lower the value of x~b(tf)].Since both the data and model depend explicitly on A, the fit proceeds over all observed wavelengths at a given time during the radiation history. As a result, ê55 and Tgb are functions of time. Since ‘Agb depends nonlinearly on Tgb, we must find the best fit values of and iteratively with the minimization constraints on To obtain starting values of ~gband Tgb for the nonlinear fit, and for comparison, we may use Wien’s approximation to IAPI(A, T) in X~b(~)’ which follows from IApl(A, T) in the limit exp(C2/AT) ~‘ 1, i.e., ‘Agb
~5.
I
I 0000
0
450-nm
‘~
0
(0
130
260
390
Time (ns)
o_i I
I
C’2
Wien —
GS
GSeI --LM -
~
b ci
40
I
I
120
200
280
Time (ns) 2 statistic for an Fe—Fe foil—LiF target impacted by a tantalum projectile traveling at 5.41 kmls, Fig. 7.2. (a) resulting in an Observed Fe foilradiation Hugoniothistory pressure and of (b) 227 x GPa and an LiF Hugoniot pressure of 122 GPa. (c) Greybody effective emissivity and (d) temperature constrained by radiation data shown in (a).
B. I
371
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
I 5t’ rWgb, — P 1 1i P \ ~ WgbJ‘t = — rWgb1AWi~/t, ~ Wgb)
‘AWgbY
— —
EWgb
1 e ~C~/ATwgt,
—,~—
The relative error incurred in approximating ‘APt by ‘AWi is equal to exp(— C 2/A T); this approximation is accurate- to within 1% for AT <3 x i0~mK (Siegel and Howell [15]). Since we can fit Wien’s relation to the data in a linear least-squares sense, we can solve for ~wgb and Twgb directly (i.e. without iteration). With these values, we may safely apply an iterative technique to (7.5) to constrain and Tgb and be assured of a nondivergent fit. We use the Golden Search (GS) and Levenberg—Marquardt (LM) iterative techniques (Press et al. [63]) to obtain three different fits: (1) GS with êg5 variable, (2) GS with Egb = 1, and (3) LM with ~gb variable. We present a greybody fit to the radiation observations from two experiments of Bass et al. [6] in figs. 7.1 and 7.2. Figure 7.1 displays a fit to data from an experiment on an Fe—Fe film—A1203 target impacted by a Ta projectile traveling at 5.67 The trend in x~b(t)suggests that thev isfitthe getsnumber better 2 ~‘ ±2Vi as km/s. the number of data get very large, where with time. Strictly speaking, x ~gb
——
I
1
0
....
—
40
\
Wien
—GS ——LM
120
200
2~
Time (ns) c ___________________________________
I
I
Wien GS GSe1 --LM —
_
a)
iT:~ITrT~’ c O
01 ~
40
dC
I 120
200
Time (ns) Fig. 7.2. (Cont.)
2~
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
372
of degrees of freedom in the fit (i.e. the number of data minus the number of parameters, 2 in this case); we might hope that X~b —2 represents a reasonable fit for the greybody model. All of the fits show Tgb(t) decreasing with time, and for the variable emissivity fits, êSb(t) increases slightly with time. This behavior is characteristic of most Fe—Fe—A1203 experiments of Bass et al. [6]. For all the Fe experiments, we note that T~~ T1(t), and that the behavior of case, from (6.1), we have jAmod(A,
t)
ê~b(t) is
consistent with assw —0. In this
EAJ(t)IAPI(A, T1(t)) ,
(7.7)
and the decrease of Tgb(t) with time (fig. 7.ld) can be explained in terms of T1(t), as detailed above. Also, the slight increase of ~g5(t)with time (fig. 7.lc) can be explained most simply by a slight decrease of the Al203 absorption coefficient upon compression. This may be consistent1with the observation that between 0.1 and 1 GPa, the refractive index of A1203 seems to decrease with pressure (——0.001 GPa Davis and Vedam [64]). Since aAUSW 0 for Al 203, this observation implies that aASW—O as well. In this case, (7.7) implies that jAmod(A, t)
êAl(0)IAP](A, T1(t)) = (1
—
rAFS)(l
—
rASF)(1
—
rAl)IAPI(A, T1(t))
(7.8)
for the Fe—Fe—A12O3 experiments. In contrast with this last fit, the experimental and greybody fit results displayed in figs. 7.2a—d, for an Fe—Fe—LiF target impacted by a Ta projectile traveling at 5.41 km/s, exhibit a relatively constant greybody temperature (fig. 7.2d) with time and a systematically decaying greybody effective emissivity with time (fig. 7.2c). In this case, Tgb(t) implies a relatively constant T1(t), as we expect for~ a2V~(Ftexp smooth 5FW interface (Grover and Urtiew [9], and eq. (5.11) above) or a reshocked interface with (fig. 5.3b) or 3FW ‘~ 2VKFtCXP (fig. 5.2b). The reshocked interface with ~FW ~° 2’S/KFtexp is less likely than the latter possibility, as implied by the data—model comparison in fig. 4.5b. The behavior of Egb(t) ê~(t)reflects a shock-induced increase in the absorption coefficient (i.e. ~ > aAUSW) of LiF via (6.6), (6.8) and (6.14). Wise and Chhabildas [16] found, via laser interferometry, that LiF remains essentially transparent up to 160 GPa. The fluctuations in the fit after about 160 ns may be due to wave reverberations or other dynamic effects, which are beyond the scope of our model, and/or possibly to the influence of reshock at the Fe—Fe foil interface, as mentioned above. In judging the value of any fit, the resolving power of the data is an issue. In particular, the ability of the data to constrain model parameters may be judged through confidence limits (e.g. Press et al. [63]). We display these for Fe—Fe—A1 2O3 and Fe—Fe—LiF experiments in figs. 7.3 and 7.4, respectively. Parts (a) and (b) represent confidence limits for x~b(50ns) and x~b(250ns), respectively. The shaded (central) region in each diagram represents that part of model parameter space which explains 68.3% of the data. Similarly, the 95.4% and 99.99% regions explain corresponding percentages of the four-wavelength data. Note that these limits are consistent with the trends in x~b(t)(figs. 7.lb and 7.2b). The basic information conveyed by these diagrams is a measure of the uncertainty of the fits; for example, from fig. 7.3, the fitted value of Tgb(SO ns) has an uncertainty of about ±400K at the 68.3% level, and about ±600K at the 95.4% level. As stated above, the interface contribution to the observed intensity dominates the SW contribution (Bass et a!. [6]). On the basis of this observation, we may reasonably fit a simplified version of the full radiation model to the data via eq. (7.5). We do this for the Fe—LiF data fit to the greybody model in fig. 7.5. First, we note that, at t = 0, ‘Amod is, from eq. (6.1),
373
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
6)
-
~Ir
3900
(‘2
5600 7300 Temperature
aL
~95.4%
.~co aa, C ii) 6) 0 Co
9000
bL I
I
68.3%
(0
96.4%
0)
(‘2
Si 5, 4) a) C) .2~
-
5400
I
683% U ~
Co ~9999%-
(‘2
I I 4500 5400 Temperature
1 3300
630u
êAI(0)IAPI(A, T1(0)) = (1
—
rAFS)
4300Temperature 5300
6300
Fig. 7.4. Confidence limits for the LM fit displayed in figs. 7.2c,d near the beginning (50 ns) and end (250 ns) of radiation histories fit by greybody model.
Fig. 7.3. Confidence limits for the LM fit displayed in figs. 7.lc,d near the beginning (50 ns) and end (250 ns) of that part of the radiation histories fit by the model. The designations 68.3%, 95.4% and 99.99% refer to that fraction of the data (which total 4) satisfied by the range of model values (~ 8b’ Tlb) encompassed within the appropriate regions and contours, subject to the assumption that the experimental errors are normally distributed.
1Amod(’~,0)
66.3% 954%
C
~9_~
3600
I I I I 3800Temperature 4600
-
3000
e°~’-’~”(1 rASF)(1 —
—
rAI)IAPI(A,
T1(0))
.
(7.9)
In this case, the magnitude of ~ A, 0) is controlled by the reflectivities, ~ and the initial value 0WF through eq. (5.10) in the simplest of T1, The which is dependent values of TF, T~and case. greybody fits inonfig.the7.2 suggest that,~TFW, for this experiment at least, T 1(t) is approximately constant. Assuming this and ~ = 0, we approximate eq. (6.1) as ~~mod(A, t) =
~AJ(t)IApl(A,T1(0)) = (1
—
rAFS)(l
—
rASF)(l
—
rAt) ea~6wh/tc0pIAPl’~ ‘A,
T1(0)).
(7.10)
In this case, the time dependence of ~ is due solely to the SW transmissivity. Using (7.10) in (7.3), along with T1(0) = Tgb(O), we may fit the Fe—LiF data for ~ We present the results of this fit in fig. 7.5. The data are cut off at 160 ns to reduce the influence of possible dynamics or Fe—Fe interface reshock on the fit. The parameter values resulting from this fit are given in table 7.1. We eliminate TAFS
B.
374
Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
E~_i I
I 0000 450-nm
0000 600-nm
i
<~~ri-~
::00ooooEo~~:
40
80
120
160
Time (ns) Fig. 7.5. Time dependent spectral radiance fits to the data displayed in fig. 7.3a for the Fe—Fe foil—LiF target. We fit eq. (7.10) to the first half of the Fe—LiF data displayed in fig. 7.3a via eq. (7.3). The slope of each continuous curve, representing the fit for the corresponding wavelength, constrains the effective normal absorption coefficient of the shocked LiF, while the intercept constrains the Fe—LiF interface and shock front effective normal reflectivities.
Table 7.1 Simplified radiation model parameters. For this fit, ~ = 4.15 mm, = 390 ns, T 1 = 4200K and rAFS = 0.08 at all wavelengths wavelength (nm)
(1— r1~~)(l — r)1)
(m’)
450
0.76
137
600 750 900
0.56 0.68 0.67
134 125 122
aASW
from the fit since it is equal to 0.08 for LiF, as estimated from n = 1.39, the index of refraction of LiF 1at STP (Weast [37]). Since thewe index of refraction seems to increase (——0.002 GPa Burnstein and Smith [65]), expect rASF ~ rAFSofforLiF LiF. Clearly, LiF haswith lost pressure some transparency upon compression. The trend in aASW toward lower values at longer wavelengths is unresolved but consistent with the Bouguer’s law expectation that a,~= 41T~~e~’/k, if we, the electromagnetic extinction coefficient (Siegel and Howell [15], p. 427) is constant or varies inversely with A. As suggested above, similar fits for Fe—Al 203 imply that aAUSW > aASW, an intriguing possibility which we do not yet understand. Lastly, we take the results of the greybody fit shown in fig. 7. id for the Fe—Fe—Al203 experiment, assume Tgb(t) = T1(t), and use (5.10) to write 1~
~ [Tgb(O)
—
Tgb(texp)].
(7.11)
~TFw = erfC(~ With Tgb(O) Tgb(texp) = 1200 K from the fit displayed in fig. 7.ld, we may calculate the trade-off between the FL—SW interface temperature due to reshock, ~TFW, and the ratio of the reshocked layer —
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
C C C
_____________________________________________________
—
—
—
—
—
—
1
—
- --
—
—
C
375
1
0.5
I
1.0
1.5
0_\VFO-
-
~i_2.0 2.0
6~/~ Fig. 7.6. Magnitude of temperature at FL—sw interface due to reshock, ~ versus the ratio of the reshocked layer thickness, ~ to the conduction length scale, ‘/~. This trade-off is constrained by the magnitude of Tgb(O) — T 65(ç5) from the greybody fit for the Fe—Fe—Al20, data displayed in fig. 7.2a. 8FW’ to the FL conduction length scale, VKFtCXP, for different values of the FL—SW interface thickness, thermal-inertia mismatch, 0~wF~These calculations, displayed in fig. 7.6, imply that the larger the reshocked layer thickness relative to the FL conduction length scale, the higher the reshock temperature at a given thermal mismatch. For this particular experiment, we expect 0~WF~ 0.1 from calculations discussed above; we also expect I.~TFws 2000 K from the calculations presented in fig. 4.5a. In this case, fig. 7.6 and model calculations imply that ~FW 2VKFtexp i0~m. Since this is a film experiment, with d — 10_6 m, we tentatively conclude that all of the film layer experienced reshock in this experiment. ~
——
8. Summary In this work we have attempted to construct a thermomechanical model for a shocked target that can be used to explain and interpret shock-induced radiation observed from such a target. We have emphasized (1) the potential effects of compression, release/reshock, and equilibrium phase transitions on the “initial” temperature profile in the target, (2) the effect of conduction on this temperature profile over the time scale of the experiment, and (3) the effect of propagation through the dynamic target geometry on radiation source intensity and its temporal variation. Comparison of the model with the results of experiments on Fe—Fe—LiF and Fe—Fe--Al 203 targets suggests the following: (1) Release / reshock calculations for Fe—Fe—Al203 targets, in comparison with the experimental results of Bass et al. [6], suggest that Fe experiences approximately 200—1500 K of reshock heating at
376
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
both Fe foil—Al203 and Fe film—A1203 interfaces when released from as245—300 GPa to interface pressures of 190—230 GPa. Below 190 GPa, reshock for Fe—A1203 interfaces appears to be minimal. Both the data and calculations suggest that the degree of reshock is strongly pressure dependent, which is consistent with the results of Urtiew and Grover [7]. In contrast, Fe released from the same range of Hugoniot pressures to Fe—LiF interface pressures between =130 and 160 GPa experiences little or no reshock. This more ideal nature of Fe—LiF interfaces is enhanced by the fact that, besides being a poorer shock impedance match to Fe than Al203, it is also a poorer thermal match, resulting in less change in the interface temperature away from the Fe release-state temperature. Comparison of data and calculations for both of these windows suggest that, while attention to the initial conditions of the interface is essential to minimize reshock, a more important factor may be the choice of window. (2) In the absence of energy sources and significant energy flux from other parts of the target, ~the 2),where = rate of change of the interface temperature,_T1(t), is proportional to exp(—~t 6Fw/2V~. For Fe at FL—SW interfaces, ~ 10 urn; consequently, a 100 urn reshocked Fe layer would relax very little, remaining near T 1(0) on the time scale of the experiment. However, if 8FW 1 um, T 6FW 1(t) relaxes almost instantaneously to its value T1(oo). T1(t) is resolvably time dependent for(3) Greybody fits to an Fe film—A1 203 experiment of Bass et al. [6] show the greybody effective emissivity, Egb (t), to increase slightly with time, while the greybody temperature, Tgb (t), decreases with time. This behavior is characteristic of most Fe—Al203 experiments. The decrease can be 8FW 2V~KFtexp for of thisTgb(t) experiment. explained in terms of the model for T1(t), and it implies that Further, assuming Tgb(t) = T the~ greybodyA fit constrains theofamount 0WF —0.1 and1(t), 8FW ~ slight decrease the A1 of reshock, ~TFw, to be s2000 K with 203 absorption coefficient upon compression can explain the slight increase of êg5(t) with time. This may be consistent with the observation that the refractive index of A12O3 seems to decrease with pressure. In contrast, greybody fits to data from an Fe—Fe foil—L1F target show a relatively constant greybody temperature and decreasing greybody emissivity. The constant greybody temperature implies a constant interface temperature, as we expect for an interface experiencing minimal reshock, while the decaying is consistent with a shock-induced increase in the absorption coefficient of LiF. Setting T1(0) = Tgb(O), we 1 (table 7.1) for fit a simplified of in thethis fullexperiment. radiation model to these data to find aASW 100 m LiF, shocked toversion 122 GPa (4) Finally, we note that the equilibrium thermodynamic estimate of the Hugoniot temperature of Fe is strongly influenced by electronic and/or anharmonic contributions to c~,at high pressure, as evidenced by both (1) the results of Boness et al. [45] when used in eqs. (4.46), (4.50) or (4.53), and (2) by requiring the solid-Fe Hugoniot and an extrapolation of the experimentally constrained Fe melting curve (Williams and Jeanloz [50]) referenced to this Hugoniot, to intersect at 245 GPa (Brown and McQueen [44]). This last constraint provides a value of ~Q(p 2, as compared to 1) = 0.046 J/kg K F( p,) = 0.090 J / kg K2 from the work of Boness et al. [45], suggesting some anharmonic contribution to c~of e-Fe. These results substantiate the arguments of Brown and McQueen [44, 22] for the importance of including electronic contributions to c~when calculating T~of shock-compressed metals. —
—~
——
—
~gb(t)
‘—
Acknowledgements We thank William W. Anderson, Mark B. Boslough, A. James Friedson, Hua Tan, Dion L. Heinz, Douglas R. Schmitt, and James A. Tyburczy for enlightening discussions. We also thank J. Michael
B. Svenolsen et al., Optical radiation from shock-compressed materials and interfaces
377
Brown and two anonymous reviewers for constructive comments on an earlier version of this work. Support from NSF grants EAR-8608249 and 8608969 is gratefully acknowledged. Contribution 4488, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125.
Appendix A. Shock front and target boundary conditions In the text we have attempted to detail the influence of certain processes, such as release and reshock, conductive and radiative transport, etc., on observed shock-induced radiation. Although microscopic processes lie at the heart of shock-induced radiation, the experimental foundation of our work requires an interpretive, conceptual framework much more abstract and general than any particular microscopic physical theory can provide. This framework is provided by continuum mechanics, which in principle enfolds and abstracts all microscopic models and viewpoints, via their appropriate macroscopic (e.g. thermodynamic) limit. With this framework, we have a representation of the phenomena partial to no particular “microphysics”, and so accessible to all. The purpose of this part of the appendix is to outline and detail the continuum framework we use to represent and interpret experimental results on shock-compressed materials. A. 1. Boundary conditions across a moving, nonmaterial surface As discussed in the text, the target is usually composed of two or three different materials sandwiched together in a plane-layer-style geometry (see fig. 2.1). The shock wave propagating through this target is a three-dimensional region with some thickness 6 (typically ~10_8 m in the materials of interest here, Kormer [1]), and moving through the material with a velocity of propagation u (typically -— io~—io~ mis). Shock compression produces large (factor of 2 to orders of magnitude) changes in any given local thermomechanical (TM) field (density) ç& over the length scale 6 and on a very short time scale (i.e., 6/~u! 10h1_1012 s), resulting in large gradients (—I~’I/6) in these TM fields across the shock front region. We assume that these length and time scales are sufficiently short so that, from the macroscopic viewpoint, they may be idealized as infinitesimal and instantaneous, respectively. In this case, we may idealize the shock front macroscopically as a moving surface S. Let the material through which S propagates occupy a spatial volume V in Euclidean space. Then S divides V into regions “ahead” (Vt) and “behind” (V) the shock front. This surface representation for the shock front is defined by the limit 6 —*0, and we note that in this limit the gradients of ~i across the shock front become infinite. Consequently, cia- loses a continuous representation in V, becoming discontinuous across S in this representation. In this case, S is referred to as singular (Truesdell and Toupin [66], section 173) with respect to ~fr,such that ——
[]as~i—~$O,
(A.1)
where [Ici’I] is the “jump” of cli across S, and ~ and cli are the limiting values of “just” ahead and behind S, respectively. In the text, we designate the states behind (—) and ahead (+) of the shock front in a more conventional fashion via the subscripts “H” and “i”, respectively. Note that S is oriented such that U(~)as u~b >0, where 1’ is the unit normal to S. The concept of a singular surface forms the basis for the continuum mechanical description of shock compression. For our purposes, we assume that S ,/j
B.
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Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
possesses no fields or properties other than a motion (i.e. u) independent of the material on either side of it in V. Assuming further that all kinematic and TM fields defined on S are homogeneous there, the
balances of mass, linear momentum, energy and entropy across S are given by [66, sections 193, 205, 241, 258] [~pU(~)j]=0,
(A.2)
[~pU(~)vj]+ [~t~]=0,
(A.3)
[IpU(~)(e + ~
v) + (v . t
—
q(~))~] as 0,
(A.4)
[lpU(~)s— T’q(~]0.
(A.5)
In (A.2)—(A.5), U~) U . 1’ is the component of the propagation velocity u as u v~of S normal to S, t o• b is the Cauchy stress vector, o~is the Cauchy stress tensor, q(~) q 1’ is the component of the heat flux normal to 5, q is the heat flux vector, e is the specific internal energy, s is the specific entropy, and T is the absolute temperature. Note that the entropy balance is an inequality via the requirement of a nonnegative entropy production on the shock front. The mass balance, eq. (A.2), is the first of the relations we use to interpret experimental results. Using (A.2) in (A.3)—(A.5), we obtain ±
p~U~)[~v~] + [~t~] = 0, + ~v.vj] +[~vt—q(~]=0, 1q(~]0 , p~U~1[~s~] [~T
—
(A.6) (A.7)
(A.8)
—
respectively. Note that (A.6)—(A.8) represent five equations in 24 unknowns: ~ U~,v~,t~,e~,q~), s~and T~. From the physical viewpoint, shock compression produces a “sudden” change in the velocity of the material in a direction normal to the shock front. In the context of the singular surface representation of the shock front, we idealize this change as a discontinuity in the component of the material velocity normal to S, i.e., [~v(~]0,
(A.9)
where 13(p) v~b. Also, note that [ju~]= 0. To substitute (A.9) into (A.6)—(A.8), we must first cast these latter relations into normal and tangential forms. To do this, we define ‘T 0 (a = 1, 2) as vectors tangent to S such that V_TIXT2I~TIXT2!,
i.e.,
~
normal
(A.10)
r~and r2 form a right-handed system. With these, we may resolve any vector into components and tangential to S. In particular, we have
V=V(V)V+V~)Ta
,
(All)
B. t_ t(j~)V+
t(T)Ta
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
379
(A.12)
.
Noting that i’~~=l,
(A.13)
V~T=O,
and assuming S is planar, that
10,
a~/3,
(A.l4) we may substitute (A.12) and (A.13) into (A.6)—(A.8) to obtain
p~U~)[Iv(~)I] + [~t(~)J] =0 PU(.?)[IV(T)I]
+
[Jt~)~] = 0
pU(~)[Ie+ ~(v~) PU(I,)[ISI]
+
,
(A.15)
,
(A.16)
v(~)~v~))I] + [~u(~)t(~) + V(~)~t~) q(~)I]= 0, —
[lT~q(~)I] 0.
—
(A.l7) (A.18)
To proceed further, we make the constitutive assumption that the material on either side of the shock front is an ideal (also known as barotropic or elastic) fluid, i.e., o——P~I.
(A.19)
With this, we have t(~)=
=
jj~
as
—P~i’•b = —P~
(A.20)
via (A.13), and = O.±.
= J/~
=
p~~’
=
0
(A.21)
from (A.14). Putting (A.20) and (A.21) into (A.15)—(A.18), we obtain
[~PI] as p~U~)[Jv(~)J],
(A.22)
p~U~)[Jv~]= 0,
(A.23)
p~U~)[~e + ~V~)I] + [~v(~)t(~)
—
PU(j,)[I5J]
—
[IT~q(~)I] 0,
q(~)I]= 0,
(A.24) (A.25)
where we have used (A.23) in (A.17) to obtain (A.24). Relation (A.22) is the second of the relations
B.
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Svendsen et al., Optical radiation from shock-compressed materials and interfaces
used to interpret experimental results. From the momentum balance, eq. (A. 15), we may write p ~U~)[~ ~v~]
+ [~v(~)t(b)~] = (v~t~ V(~)t(~)) + ~[~v(~)t(~]. ~
—
(A.26)
Putting this into the energy relation, eq. (A.24), we obtain p~U~)[~e~] = ~(v~)t)
V~)t~))
-
-
~[~v(~)t(~]
+
[~q(~]
-
~[~v(~](t~) + t~))+ [~q(~].
(A.27)
From the mass balance, we have =
~
[jp~]U~
(A.28)
.
Putting this into (A.27), we obtain
[~e~]= ~
~±
[~p~](t~ + t~))+
[~q(~]
=
-~[~1Ip~](P+ P~)+ PU~) [~q(~],
(A.29) where we have used the ideal fluid constitutive assumption (A. 19). Finally, if we make the constitutive assumption that the shock front is adiabatic, i.e., [~q(~] = 0, (A.29) reduces to ~(P
[~e~]=
+
Pt),
(A.30)
with ‘q 1 p ~ - being the relative compression. Relation (A.30), known as the Rankine—Hugoniot relation (e.g. Rice et al. [20]), is the third relation used to interpret experimental shock compression data. With the adiabaticity assumption, (A.25) reduces to —
p~U~)[~s~] [~1/T~]q~) .
(A.31)
If we further assume no heat flow, i.e., q~)= 0, (A.31) reduces to
[Isl]
0,
(A.32)
implying that the specific entropy must increase across an adiabatic shock front separating two non-heat-conducting, ideal fluids, as previously established by other means (e.g. Bethe [67]). Relations (A.2), (A.22) and (A.30) represent the balances of mass, momentum and energy, respectively, used in this work to interpret experimental shock compression results, and for various related calculations of Hugoniot, or shock-compressed, states. Along with (A.32), they constitute a thermodynamic description for the “experimental” shock front.
A.2. Target boundary conditions The shock front represents a kind of boundary in the target, across which we may use, given the necessary experimental or other information, (A .2), (A.22) and (A.30) to calculate the change in
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
381
density, material velocity, pressure, specific internal energy, etc., during shock compression. As stated above, however, different materials make up the target, introducing a further “discontinuity” into the field description of the target as a whole. Consequently, we must also find the balance of mass, momentum, energy and entropy across the boundaries between target layers. Since we assume that, as each target layer is compressed and the shock front passes on into the next target layer during the experiment, the layers do not separate or blow apart at their interface, this boundary may be regarded as a material surface, such that U(b) = V~), or U~1= 0. This states that, at a material surface or interface, the surface moves with the displacement velocity of the material on either side of it. Consequently, at such an interface [Jv(~] = 0, and the balance of momentum becomes [~t(~)~]=0,
[~t~,.)~]=0,
(A.33)
from (A.15) and (A.16), respectively. Similarly, the balance of energy, eq. (A.17), reduces to [~V(~.)a, ]t
= [~q(i,)
],
(A.34)
where we have used (A.33) in (A. 17) as well. Under these circumstances, the entropy inequality (A. 18) simplifies to [~T~q~]~
0.
(A.35)
Substituting the constitutive assumption (A. 19) into (A.33), the balance of momentum at a material interface in the target takes the form [~P~]= 0.
(A.36)
Under the constitutive assumption (A.19), we also see from (A.21) that t~ are identically zero, and with this the balance of energy (A.34) then requires that the normal component of the heat flux be continuous across the interface, i.e., [~q(~]
=
0.
(A.37)
Substituting (A.37) into the entropy balance (A.35), we have q(~)[ITU0, implying that the interface between the two ideal fluids will achieve thermal equilibrium (i.e. no entropy production) if (1) either ideal fluid (and so both) is adiabatic (q~)as 0) and/or (2) temperature is continuous across the interface, i.e., [I T~]=
0.
(A.38)
We assume (A.38) in all energy transport models presented in this work (see appendices D and E). Relations (A.36)—(A.38) then represent boundary conditions appropriate for the assumed constitutive nature of each target component, and consequently are consistent with the analogous shock front relations given above.
B. Svendsen
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et al., Optical radiation from shock-compressed materials and interfaces
A.3. Shock velocity—material velocity relation
Beyond the basic assumptions discussed in part A.1 underlying the balances of mass (A.2), momentum (A.22), and energy (A.30), the basis of the relations between the Hugoniot and isentropic properties is the constitutive assumption that the shock front propagation velocity U1 may be written as
[~vI] as
a function of as
u
=
—
v~in a Taylor series about v1. To second order, this is
—
+2o([~v~]~). a1 + b1[~v~]+ c~[~v~]
(A.39)
The coefficients in (A.39) are defined by a~ U~~
11511..0,
(A.40)
b~as dU1/d[~v~]~[1U1]..0,
(A.41)
c~
~
~
(A.42)
.
Now, using the relative compression, ~ = 1 form
—
P1/PH’
as
defined in the text, we may write (A.2) in the
[IvI]=,1U1.
(A.43)
Following Prieto and Renero [68], we now substitute (A.43) into (A.39), rearrange, and square both sides of the result to obtain 4U~[(1 b 2 2a~c 2]U~ + a~= 0, (A.44) c~rj 1?)) 11) i.e., a biquadratic relation for U 1 in(A.44) terms has of a1, c1 and As shown by Prieto and Renero [68],for 2/4a?)2, theb1,solution b1’q <1 and c1 (1 b,fl) U~= a~/{(l b 2 2a 2}. (A.45) 1rj) 1c1?) —
—
—
~.
~
—
—
—
Substituting (A.43) into (A.22), we have either U~= (1/p~?))[~P~],
(A.46)
or [IvI]2= (?)ip
1)[~P~],
(A.47)
depending on whether or not we eliminate [~v] or U1, respectively. Putting (A.45) into (A.46), we obtain 2—2a 2} (A.48) [~PI] = p~a~~/{(1 b1~) 1c~~ Note that c 1, when resolvable by experimental data, is usually negative (e.g. Pastine and Piacesi [69], .
—
B. Svendsen a a!., Optical radiation from shock-compressed materials and interfaces
383
Ruoff [40], Prieto and Renero [68],Brown and McQueen [44]). In this case, we see that PH(cI = 0) as b1?)—* 1, but PH(cl <0)—* p1a1b~ic1in this same limit (Prieto and Renero [68]). Clearly, c1 cannot —*
equal zero for physically reasonable asymptotic behavior. Setting c~= 0 in (A.48) reduces it to the so-called shock wave “equation of state” (McQueen et al. [21]). For the r and/or other high-pressure phases of Fe, for example, c1 is apparently unresolvable by implication of how well the uv~ data are fit by a linear relation (Brown and McQueen [22]). Following Pastine and Piacesi [69], we substitute (A.46) into (A.40) to obtain 2~ a1 = (p[~P~]i~)’’ 115110 (A.49) .
Since the limits becomes
PH
~ p1, q —*0 and [~P~]—*0 are formally and physically equivalent to [~v] —*0, (B .35)
a1=0/0, an indeterminate form. However, since both [~P~]and ~ are analytic functions of ~ or PH via (A.48) and by definition, respectively, we may use L’Hopital’s rule to evaluate (A.49), i.e., 112 a.=~— , (A.50) (1 (d[~P~]/dpH)~ \p 1 (di~/dp~)/ PHPI or (A.51)
112~~P. a1 = (BHipH)
Using (A.46), (A.47), and their derivatives, we find, in a similar fashion, an expression for b~,i.e., b 1 = ~[1 + B~IpH~~p.],
(A.52) (A.53)
1p~(d2PHidp~) + 1. B~as(dBHidPH)= B~
Again in a similar fashion, we obtain 1 =
1 b 1(2 b1) + 2BH/dP~),
~—
—
as
~-~--—
(BHB~(,~P,
(A.54) (A.55)
(d 1p~(d3P
B 11B~= B~
11/dp~)(B~ 2)(B~ 1). —
—
—
(A.56)
Note that, up to this point, nothing specific has been assumed about the “initial” state ~ as { T1, P~,u~p1, s~,e1} to which (A.39), and so a1, b1, and c1, are referenced. If values for a~,b~and c1 are constrained by experiment, for example, then is the state of the uncompressed starting material, usually with T1 as 298 K, P1 = 0.1 MPa, and v1 = 0. But, as we have done in the text to write (4.19)—(4.22) and the relations that follow, we can use equivalent thermodynamic paths to “shock compress” the material from any initial or reference state. .~,
384
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Bethe [67] has shown that, for shock compression of a non-heat-conducting ideal fluid, i.e. that represented by (A.2), (A.22), and (A.30), the initial (PH = p1) slope and curvature of the shock compression locus are equal to those of an isentrope referenced to the same initial state, i.e., BH!~P = K2,
(A.57)
~
(A.58)
as
K~,
where K2 and K~are the isentropic bulk modulus and its first pressure derivative referenced to (4.25) in the text, we may also relate B~to K’ (e.g. Pastine and Piacesi [69]), i.e., (BHB~)~PP = K2K~+ 2b1’y1
.~.
Via
(A.59)
.
In these relations, K2, K and K’ are the isentropic bulk modulus, and its first and second pressure derivatives at constant entropy. Note that K2, K~and K’~are equilibrium thermodynamic properties. In principle, (A.57)—(A.59) may be used as consistency relations between Hugoniot data and other types of compression data for appropriate materials. Substituting (A.57) and (A.58) into (A.51) and (A.52), respectively, we obtain ,2 (A.60) a1 = (K2/p~)’~ b
(A.61)
1=~(l+K~).
Doing the same with (A.59) in (A.55), we have c~=
~—
[2b~(2
—
b1
+
y1)
+
K,K~].
(A.62)
From (A.61) and (A.62), we see that, if c~as 0, then K,K’ as ~ (1 + K~)(K~ —7— 4~),
(A.63)
and this is the case for most materials from the experimental resolution point of view. We presume that values a’ and b~for a u—va relation have been constrained by experiment such that the shock wave “equation of state” (4.10) describes the (RH, PH) states of the high-pressure phase ~ along the Hugoniot. These states are also described by the “metastable” shock wave equation of state (4.23) based on ar and hr. Consequently, we may substitute (4.10) into (A.51) and (A.52) referenced to . r to obtain 3]112a~’, (A.64) ar as (1— 4.)[(1 + b~t~)/(1 b4) b~—1(1+ 4(1—4)b~+(2— ~)(b~)2~—1~ (A65
pr~
—
—
4 ‘\
respectively, where 4
(1 1
—
—
b~j)(1+ b~q5)
‘
p~’pr.Note that, from these relations, ar—> a 1 and br—+ b1 as p11—* p~,i.e.,
385
B. Svenda-en et a!., Optical radiation from shock-compressed materials and interfaces
as 4—*0. Note that we need p~to obtain ar and br, and so Kr, Kr’, and Kr” with cr=O, our initial goal. In the text, for example, we use relations analogous to (A.64) and (A.65) to estimate the isentropic properties of liquid Fe referenced to STP given in table 4.1. A.4. Impedance match
]
The balance of momentum at the boundary between adjacent target components, [~ = 0 and (A. 36), combined with the balance of momentum (A.22) across the shock front, form the basis for the impedance match technique (Rice et al. [20]), which is used to calculate the density, pressure, etc., of the high-pressure, shock-compressed states of each target component throughout this work. For completeness, we present here a brief development of the impedance match relations between any two adjacent target component layers A and B in the target. We orient these layers by having the shock wave propagate from A into B. The basis of the impedance match is found in the idea that the shock wave propagating from A into B accelerates, but does not shock, A material next to (i.e. within approximately a shock front’s thickness of the interface) the interface to “impact” with the B material adjacent to the interface. The compression of the interface then shocks A material next to the interface, and sends a shock wave propagating into B material. Imagining this compression takes place from zero pressure, the impact of A against B occurs at a velocity twice that of the material velocity of A, UHA in what follows). In the process, A material next to the interface (note that 13H U(j,) and v. as achieves the pressure, ~HB’ and normal material velocity, VHB, of B in order to maintain a material interface. This requires a net change of 2~HA 13HB in the material velocity of A interface material; this change is then the material velocity imparted to A material next to the interface by the shock wave. Again, we emphasize that A interface material is not shocked by the shock wave propagating through A; rather, it is shocked as it “impacts” B. Hence, the material in A adjacent to B at the interface between these layers experiences a different shock compression history than the material in the interior. Since we do not measure the shock wave velocity in the radiation experiments, the basic aim of the impedance match for our purposes is to calculate the normal component of the shock-induced material velocity in B as a function of (1) that in A, and (2) certain material properties of each layer, as the shock wave propagates from A to B. As emphasized above, in this impedance match, we are dealing with the A layer material “right” at the interface, which is not shocked by the shock wave propagating through the A layer, and its analog in B. To begin, we have from (A.22) the pressure generated in A interface material and B material as a result of shock compression, i.e., —
~HA = ‘~iA+
PIA(uA
~HH = 1~iB+
PIB(UB
—
UIA)(vHA
— ~IB)(~HB
(A.66)
— ~
—
(A.67)
~1B)
(with P 1 as p and ~H as P ). Again, we emphasize that (A. 66) represents the shock-compressed state of A interface material after it has impacted B interface material; this state is in general different from the shock-compressed state of A material in the back of the interface. To proceed further, we adopt the constitutive assumption (A.39) for both layers, 13iA = alA + blA(vHA CIA) + clA(UHA ~1A) , (A.68) 2 +
—
—
—
u~ 3= ~
+
bIBVHB
+
cIBUHB.
(A.69)
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B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
In (A.69), we have used the fact that ~ 0 in the experiments. From the discussion above we have = 0 when we require VHA = 13HB for a material interface. Putting that 13HA 13iA = 2VHA VHB, or this along with (A.68) into (A.67), and assuming =0 = P 1~,we obtain —
—
13iA
~jA
2vHA
—
~HB) + PIAbIA(2VHA
—
VHB)2
(A.70)
+ plAcIA(2vHA — vHB).
plAaIA(
~HA
Likewise, substituting (A.69) into (A.67), we have = PIBaIB13HB + PIBbIBVHB + pIBcIBvHB
~HB
And since
=
~HA
~HB
plAaIA(213HA
—
(A.7l)
.
for a material interface, these last two expressions combine to give ~HB) + PIAbIA(2VHA
—
vHB) + PIACIA(2VHA
= pIBaIBvHB + PIBbIBVHB + pIBcIBvHB
Rearranging this into a polynomial in
VHB,
—
~HB)
(A.72)
.
we obtain (A.73)
UHB+aUHB+bvHB+c=O,
a
as (PIAbIA
b as 12p1Ac1Av~A
PIBbIB) — 6pIAcIAvHA PIAcIA — PIBcIB —
2p~~(a~ + 2bIA13HA PIAcIA
—
—
(PIAaIA + 4PIAbIAUHA + plBalB) PIAcIA — plBclB
+ 4cIAv~A)vHA
plBclB
We may solve (A.73) for 13HB by first substituting 13HB = w ~a into (A.73) to get w3 3pw 2q = 0, with p as ~(a2 3b) and q ~(9ab 27c 2a3). Next, we set w = \/75 cos 0 and use the trigonometric identity 4 cos30 3 cos 0 = cos 30 to obtain cos 30 as q/p~./~,and so 0 as ~ cos’(q/pV~). Since 30, 30 + 2ir, and 30 + 4ir all have the same cosine value, we have w = 2~/~x {cos 0, cos(0 + 21T/3), cos(0 + 41T/3)}, i.e. three roots for w. Consequently, we obtain —
—
—
—
—
—
—
13H5
=2-s/J5X
{cos0,cos(0 +21T/3),cos(0 +4ir/3)}
—
(A.74)
~a,
representing three roots, i.e. possible values, for VHB. Little of a general nature may be said regarding this solution. The only general guideline we have to choose the appropriate root is the physical requirement that 0HB be nonnegative real, just as the coefficients a, b, and c are real. Beyond that, we must use physical plausibility to make this choice. For the particular purpose of the experiments, we are usually justified in assuming PIAcIA as PIBCIB. In this case, (A.72) reduces to
(A.75)
VHB+avHB+b=O, =
a
—
12pIAcIAv~A
—
(PIAbIA
(PIAaIA + 4PIAbIAUHA + PIBaIB) — PIBbIB) — 6PIACIAUHA
b
+ 2bIAVHA + 4cIAv~A)vHA (PIAbIA — PIBbIB) — 6pIAcIAvHA
= 2P~A —
B.
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
387
and solutions (A.76)
VHB±—2(a±Va4b).
Since a and b are, in general, real, we require a2
4b in order that
addition, we require VHB_*O as UHA~_*O,which gives us 131-IB
13HB
be nonnegative real. In
as 13HB~i.e.,
—~(a—Va2—4b),
(A.77)
since b—*0 as Using (A.77), we now write the impedance match for the particular type of targets used in the experimental studies presented in this work. These consist of (1) a metal driver plate and (2) one or more “sample” layers (e.g. the film/foil and window layers in fig. 2.1). The driver plate is usually impacted by a metal flyer plate traveling at a velocity Vim~Letting the flyer plate be layer A, and the driver plate be layer B, we have, from (A.77), (A.78) where now a=
—
plFalF
+ 2PiFbiFVjm + PIDaID
PIFbIF
with
2~HF
=
Vim
—
b
= PIF(aIF + bjFVjm) Vim
PiD’~iD
PIFb~F
—
PIDbID
and c
1~= 0. In these expressions, the subscripts “F”, “D”, and “S” stand for the flyer plate, driver plate and sample layers, respectively. For any two layers in the target, such as between the driver plate and sample layer, we have (with c1~~ = 0)
(A.79)
2—4b), VHS=—~(a—Va a=
—
p~DaID+ 4PIDbIDVHD + p~ PIDbID p~~b~ 5a1~
2P~~’~~D + 2bIDVHD)UHD
1-’ts 5 PIDbID P~s for the normal material velocity in the sample layer. The impedance match between any remaining layers has this form, in the linear approximation. So, given the impact velocity, 131m’ of any experiment, and the appropriate material parameters of each layer, we may calculate the material velocity achieved in each layer due to shock compression as constrained by the requirement of material interfaces between layers. From these velocities follow the shock wave velocity via the constitutive assumption of a shock velocity, material velocity relation [e.g. (A.39)], and pressure via (A.22), of the first shock state achieved in each target layer. —
b
=
—
Appendix B. Equilibrium thermodynamics and shock compression The calculations discussed in the text for Fe—Fe—Al 203 and Fe—Fe—LiF targets are based on equilibrium thermodynamic models for the Helmholtz free energy, F( T, p), of the solid and liquid states of Fe. As discussed in the text, this model is based on (1) a Debye model for the harmonic
388
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
contribution, (2) the low-temperature (T ~ TF, the Fermi temperature) electronic contribution, and (3) high-temperature (T> TD, the Debye temperature) anharmonic contribution, which we combine for simplicity, since they are of the same order in T. The liquid state model is presented elsewhere (Svendsen et al. [48]). In the last two parts of this appendix, we detail certain relationships between isentropic and Hugoniot states using this model, and a method to recenter experimental U—V relations to the STP density of the high-pressure phase, both of which are used in the calculations. B.1. Solid-state equilibrium thermodynamic model For a cubic or isotropic solid material subject to an isotropic state of stress, the combination of the Debye model for the harmonic contribution along with the high-temperature anharmonic and lowtemperature electronic contributions provides an expression for the Helmholtz free energy, F( T, p) (e.g. Wallace [46] sections 5, 19, 24; eq. (4.33) in the text), i.e., F(T, p)
as[~1(p)
+
~yRT~(p)] + 3vR[ln(1
—
e~t))
—
~ED(~D)]T
+
A2(p)T2
—
~F(p)T2.
(B.1)
Note that F has units of J / mol. For simplicity, we neglect possible bandstructure and electron—phonon interaction contributions to F in writing (B.1). Relation (B.1) is correct to 0( T 3) in the anharmonic contribution to F, and to all orders for the harmonic contribution in the context of the Debye approximation. This slight inconsistency is due to the unavailability of a tractable anharmonic model, analogous to the Debye model, for the materials of interest. Also, for Fe, we are guided by the results of Andrews [42], who was able to fit various data on the thermostatic properties of the a- and c-phases with a Debye model for the harmonic contribution to F, ignoring anharmonicity altogether (although anharmonicity may be reflected in the value of his F). As discussed in the text, the anharmonic coefficient A 2 ( p) is dependent on the particular model chosen for the pressure and temperature dependence of the anharmonic phonon frequency spectrum. Since we do not have such a model for the materials of interest here, we simply combine it with the electronic contribution to form Ii(p), as given in the text. On this basis, (B.1) provides the means to a rational parameterization of the approximate density and temperature dependence of a number of solid-state properties, such as the pressure, i.e., 2, P(T, p)as p(9F/ô ln P)T P(0, p) + 3vRpyDED(~D)T+ ~pwQT P(0,p)asp(dtP/dlnp)+ ~VRPTDYD, where
YD
(B.2) (B.3)
is the lattice Grüneisen parameter in the Debye approximation, as defined in the text. From
(B.l), the molar entropy is given by S(T, p)as —(c9F/ôT)~ ‘3vR[~ED(~D)—ln(l —e~)] + LiT.
(B.4)
The isothermal bulk modulus is given by KT(T,
-) as
~
e~~lT+ ~pw(l—w)QT2, (B.5)
B. K(0,
P(0, p)
p)as
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
+
389
p[p(d~I3idp) + p2(d2a~Pidp2)]+ ~ziRpyDTD(yDqD), —
q~as_dlny~/dlnp,
(B.6)
which is assumed constant. The molar heat capacity at constant volume (density) is CV(T,p)as_T(ô2F/ôT2)V_3VR(4ED(~D)_ ~
(B.7)
1)+LiT.
The change in pressure with temperature at constant density is given by, from eq. (B .3), pycv
(~Pi~T)~ = aKT
=
=
3v(R/M)PYD(4ED( ~D)
—
e~-~ 1)
+
pw([liM)T,
(B.8)
with c~= C~/M.From (B.7) and (B.8), we have ycv = yDcv +
(w
—
(B.9)
YD)(L/M)T,
and so Y
7r
+
(w
—
(B.10)
YD)(Li~V)T,
which is very weakly temperature dependent above TD since w as y~and 12 ~ 3 yR. Other properties given by a ratio of the derivatives of F(T, p) include the coefficient of thermal expansion, a(T, p)
=
pyc~iK~,
(B.1l)
the isentropic bulk modulus K2(T, P) = KT
+
(B.12)
pyc~yT,
and the specific heat capacity at constant pressure, (B.l3)
c~(T,p)=c~(1+ayT). These are the solid-state model properties used in the text. B.2. Connection of isentrope and Hugoniot
To use the model as detailed above, we need to calculate the change of certain model properties, such as P, with density at zero temperature. So, we relate the zero-temperature model properties to the known change in density along an appropriate isentrope or the Hugoniot, as follows. From (B .3), we have ~
with
~Ds as
2PWQTSH,
TD/TSH. P(TSH, p) is
given
by eq. (4.32) in the text, i.e.,
(B.14)
390
B. Svendsen a a!., Optical radiation from shock-compressed materials and interfaces
P(TSH, PH) =
Pj
+
PHYH AeSH + (PH/P~)[l
—
(l +
27H)?)ls](1’H
—
P1),
(B.15)
and 1~eSHis calculated numerically, as discussed in the text. In a similar fashion, we have, from (B .5) and (B.12), 3’YD)ED(~D)T
KT(O,
p) = K2(T, p) p-yc~T 3v(R/M)pyD(l qD —9v(R/M)pyD e~~lT— ~p(l—w)w(12/M)T2. —
—
—
—
(B.16)
To get an expression for K 2(T, p) along the Hugoniot, we follow McQueen et al. [21] in equating an infinitesimal change in specific internal energy (SIE) along the Hugoniot with one along an equilibrium thermodynamic path, as follows. From the first law of equilibrium thermodynamics, we may write, at a given density Pr’ the relation e(s, Pr) = e(sr, Pr) +
f
T(s*, Pr) ds*,
(B.17)
where e(s, p) is the SIE and s is the specific entropy. Since we have assumed the shock-compressed state is one of thermodynamic equilibrium, we may set Pr = PH’ Sr = SH, and write de(s, PH)
=
de(sH, PH) + d(J T(s*, PH) ds*),
(B.18)
giving us an expression relating an infinitesimal change in SIE along the Hugoniot, de(sH, PH)’ to one at the same density but at another specific entropy. From eq. (4.21) in the text, we have another expression for de(sH, PH), i.e., (B.19) PH
Substituting this into (B .18), we have ~
T(S*,pH)ds*).
(B.20)
Now assuming s = s(P, p) [i.e. e = e(P, p)], we have 1 K Tds=—dP_--~-dp. P7 P7
(B.21)
Putting this into (B.20) and rearranging, we obtain K~=
7H~— ~7H(~H
+
Pr) +
p1
[1—(1+ ~yH)?)p]BH YHP~~ PH —
(J
T(s*, PH) ds*),
(B.22)
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
391
BHaspfld[~PI]/dpH=(1 —n~)dPH/d?)~, where YH
as
7(~H,PH). Letting P
KS(TH, PH)=(PH/Pr)[1
—*
~H
and ~
--*
(B.23)
~I1’
we obtain the desired expression,
—(1+ 27H)?)It]BH+ 27H(”H
P1).
(B.24)
In the text, we further assume YH as Thu’ i.e., that y is a function of density alone.
Appendix C. Isentropic release and reshock Considering the shock-compressed material as a thermodynamic “system” and the “lab” as its “surroundings”, a balance of energy implies that any infinitesimal change in the specific internal energy of the system, de( q, w), is due to the difference between the net amount of heat transported into the system from the surroundings, q, and the net amount of work done by the system on its surroundings, w. If we assume the material to be internally in thermodynamic equilibrium, then de = T ds P dU and any infinitesimal change in specific entropy of the system is given by —
(C.1)
ds=~q—~(w—PdV).
Adopting the idea that heat transport in or out of the target is insignificant on the time scale of release, we assume that the release path is adiabatic (q = 0). Further, we assume that the mechanical work done by the system during release is entirely reversible (w = P dv). In this case, the release path is both isentropic and adiabatic. If a phase transition occurs during release, the constraint of isentropic release in turn places constraints on the phase transition. Consider a first-order transition from the shock-compressed phase 13 to a release phase ~r.In this case, the total specific internal energy, e, total specific entropy, s, and total specific volume, v,. of the two-phase system may be written in the form (C.2) where x is the mass fraction of ir, and 4i as {e, s, v}. As with a single-component system, an infinitesimal change in specific internal energy of the two-phase system, de, is balanced by the net heat flow into the system from the surroundings, q, minus the net work done by the system on its surroundings, w. Assuming that each phase is internally in thermodynamic equilibrium (i.e., temperature, pressure and composition are homogeneous within the phase), an infinitesimal change in total specific entropy of the two-phase, single-component system is, from (C. 1) and (C.2), T~ds = q + ~(T~
—
T’~)ds” + [(gP g~)+ —
s1T(T~—
T”)] d~
—
Wjr~
(C.3)
where P~d[(1
T]
x)u~] P d[XV’ is the nonrecoverable work done by the system on the surroundings, and g is the specific Gibbs free Wir as
—
—
—
B.
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Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
energy. The second term on the right-hand side of (C.3) is the entropy produced by heat flow between phases, while the third term is that produced by mass exchange between phases. A sufficient, but not necessary, set of conditions for isentropic (ds = 0) release through the mixed-phase region is then, on the basis of (C.3), (1) adiabatic release (q as 0), (2) mechanical equilibrium (P~= P”) in the system plus reversible (P dV) work on the surroundings (P = as P’~),(3) internal thermal (T~= T”), and (4) chemical (g6 = g’~)equilibrium. Clearly, from (C.3) we see that even if release is isentropic through the mixed-phase region, it is no guarantee that the phase transition will occur in thermodynamic (i.e. thermal, mechanical and chemical) equilibrium. We choose to satisfy the constraint of isentropic release by assuming the conditions discussed above consistent with this, i.e. (1) release is adiabatic, (2) all work is recoverable, and (3) the phase transition occurs in thermodynamic equilibrium. On this basis, we need an expression for the change in pressure and temperature across an isentropic phase transition. With s = s(P, T) for each phase, an infinitesimal change of s in the mixed-phase region is, from (C.2) with 4 = ds = (c~+ x Ac~)dlnT— (a~V6+ x Aav)dP
+
As d~,
(C.4)
where c~is the specific heat at constant pressure, V 15 the specific volume, a is the thermal expansion, and A4 as ~ is the jump of any quantity 4 across the mixed-phase region. For an isentrope through the mixed-phase region, ds = 0. Putting this condition into (C. 4), and noting that pressure and temperature do not vary independently along the equilibrium phase boundary, i.e., At dP as As d TPB, we have —
[(c~ + ,y Ac~)AV—(a6V~+ x Aav)TPB As]dP
+
As TPB ~ d,~k’= 0.
(C.5)
We now propose to solve (C.5) for x and apply it to the problem of isentropic release when the release path traverses all or part of a mixed phase region, as discussed in section 4.2 in the text. Let 11(P, x)
as
(c~,+ x Ace) Av
X(P)
As TPB As.
—
Tp~(a~V~ + x Aav) As,
(C.6) (C.7)
Then (C.5) may be written H(P,~)dP+X(P)d~=0.
From (C.6) and (C.7) we have
HX(P, x) as (aHh)x)~= Ac,, Av X,,(P)
as
—
TPB
Aav As,
(oXioP)~= As (dTPB/dP) As
+
2 As TPB(dAs/dP)PB
(C.8) =
[Au + 2TPB(dAs/dP)PB] As. (C.9)
Note that both of these partial derivatives are functions only of pressure along the phase boundary.
)
B.
Svendsen et a!., Optical radiation from shock-compressed materiaLs and interfaces
393
Since at any point along the phase boundary P as P~= P~,we have (dAs\ ~—J \ dP /PB
Ac~Au ——--Aav. TPB As
(C.10)
Putting this into (C.9), we have
X,,(P) = (AU
—
2TPB Aau) As
+
2 Ac~AU.
(C.11)
Now, (C.5) will be an exact differential if X,., as lix. Since, from (C.8) and (C.1l), this is clearly not true, (C.5) is not an exact differential equation, in its current form. However, we may attempt to put it into such a form by solving pii(P, x) dP + p~X(P)d~ Ø,
(C.l2)
where p~is the integrating factor. Putting these into the exactness criterion, we require (~H)~ as (~X)~.
(C.13)
Expanding (C.12), we obtain
H/~~XILp+(H~Xp)I20.
(C.14)
Note that (H~ X~)is a function of P only. In this case, we set p~= 0 and solve —
(H-X ‘~
X
~,
(C.15)
to obtain the integrating factor,
~(P) With
as
p~P),
exp{f
~-
[AaV
-
TPB
+
j]
dP}.
(C.16)
(C.5) in the form (C.12) is now exact. On this basis, we may solve (C.5) as follows. Define
V’~(P,x) as ~(P)H(P, x),
(C.17)
~
(C.18)
x)
as
p~(P)X(P).
Integrating (C.17) with respect to pressure at constant composition, we obtain
~(P, X)J~(P*)H(P*,X)dP* + g(x), where g(,~’)is at most an arbitrary function of composition. Putting this into (C. 18), we have
(C.l~)
B.
394
~(P)
~(P)X(P)
Svend.sen et a!., Optical radiation from shock-compressed materials and interfaces
=
J
~(P*)~(P*,
x) dP* + g~(x).
(C.20)
Solving this for g~ ( x), we find
I
g~(x)= ~(P)X(P) ~(P*)H(P* x) dP*. Even though it appears that g(,y) is a function of —
pressure,
(C.21) it cannot be, by its definition (C.20).
Further, since HX(P,
x)
=
Ac~AU
(C.22)
TPB Aav As
—
is actually not a function of x~we have
g(x)—xg~(x),
(C.23)
and so
~(P, x)
=
~~(P)X(P)
+
J
~(P~)[H(P~ x)
—
Xhjx(P*)] dP*,
(C.24)
with H—H~=c~Au— TPBa~u~As,
(C.25)
from (C.7) and (C.22). Putting (C.25) into (C.24), we obtain 1I~(P,x) =
1(P)
as
J
~p~(P)X(P)
~(P*)[T
Note that ‘I’(P, x)
is
—
1(P),
a~u~ As
—
(C.26)
c~AV] dP*.
equal to some constant since
~P~dX+~1tpdP0
(C.27)
by definition (C.12) above. Consequently, (C.26) gives us ~1’(P,x) = ~~(P)X(P)
—
1(P)
=
constant.
(C.28)
As an initial condition, we assume that the mass fraction of ‘rr in the Hugoniot state is XH as x(PH). In this case, (C.26) gives us
~P(P,x) as W(PH, XH)
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
395
or
~(P)
=
~1
~(P)X(P) (~(PH)x(PH)xH
~(P*)(T
aV
As
—
c~AU) dP*)
(C.29)
This last relation is used in section 4.2 to estimate the effect of possible melting transitions on the release process.
Appendix D. Energy transport in the target In this appendix, we (1) derive conditions under which radiation and/or conduction may be an important means of energy transport in the shock-compressed or released states of the target components, and (2) establish energy balance relations for the target components to be used in appendix E and the text. We attempt (1) via a dimensional analysis of the local energy balance in a radiating, conducting target represented as a rectangular Cartesian continuum with material properties assumed to be isotropic and homogeneous in the reference (deformed) state. In constructing this energy balance, we assume that the radiant energy density and radiation stresses are negligible in comparison with the corresponding thermomechanical quantities. We also assume an equilibrium caloric constitutive relation for the specific internal energy of the relevant deformed state of each target component. With the deformed state as the reference state, then, we may assume, for example, that the heat flux q is given by the classic Fourier relation in an isotropic medium qas—kVT,
(D.1)
where k is the thermal conductivity. Under these circumstances, the local balance of energy in the neighborhood of the (deformed) reference state r is represented by Prcpr
da-T(x,
t) =
kr V2T(x, t)
—
div h(x, t)
+ p(x, t).
(D.2)
In this relation, T is the absolute temperature, h is the radiant energy flux, and we have combined terms for the specific internal energy supply and mechanical dissipation rates to form ~(x, t). Also, Pr’ cp,. and kr are the density, specific heat at constant pressure and thermal conductivity of the reference state. Note that, in (D.2), time is judged relative to the beginning of the reference state, and s9a- as 81ôt. Of the terms in (D.2), neither div h, the divergence of the radiative flux, nor ~, the specific internal energy production rate, is specified. To find an expression for div h, we first need to discuss radiative transport in the target. All radiative transport models discussed in this paper are founded in geometric optics and the classical equation of transport [eq. (D.3) below], both of which assume an optically isotropic propagation medium (i.e. one with a uniform index of refraction). The form of this equation we use here presumes that (1) all processes affecting the observed radiation intensity (scattering, absorption, etc.) are independent of the intensity of the sources and (2) any scattering is elastic and isotropic. Under these conditions, the change in the quasistatic spectral intensity, ii(s), at a point p(x) in the direction s is given by (e.g. Siegel and Howell [15], eq. (14-4))
B.
396
Svendsen et a!.. Optical radiation from shock-compressed materials and interfaces
f
~
i~(w~)dw~.
(D.3)
This is the radiant energy transport at a point p(x) per unit time, per unit projected surface area normal to s, per unit wavelength interval dA about a single wavelength A, per unit solid angle in a single direction s. Note that the subscript A denotes a spectral quantity. i is quasistatic because we assume it is not explicitly dependent on time, but only implicitly so through temperature, etc. In (D.3), V as ~ 0~A= o~ (A, T, P) are the spectral absorption and scattering coefficients, respeca 5 as a5and (A, T, and tively, ~e P) = ~a(A, T) is the spectral emission intensity, all along s. The first term on the right-hand side of (D.3) is the loss of intensity by absorption (including the negative contribution from induced emission), the second is the gain by all emission processes except induced emission, the third is the loss of intensity by scattering, and the last is the intensity gained by scattering into the s direction from the solid angle w. Note that we assume, in writing (D.3), that the wavelength of the radiation is not changed as a result of these processes. If we integrate (D.3) over all solid angles and wavelengths, assuming any scattering to be isotropic, we obtain div h at a point p(x) (e.g. Siegel and Howell [15]), i.e., div h
=
4~Ja5(A )[i(A)
—
~(A*)] dA*.
(D.4)
In this relation, the first term represents the rate of emission of radiation per unit volume in all directions, while the second, given by
f
~(A)as
i~(A,w*)dw*,
a =4,r
is the radiation intensity scattered from all directions into p at wavelength A. In principle, we could substitute (D.4) into (D.2) and try to solve the resulting nonlinear integrodifferential equation for the temperature field in the medium of interest. However, here we want only to establish the magnitude of div h relative to other terms in the energy balance, eq. (D.2). To judge the relative magnitude of the terms constituting (D.2), we render them nondimensional (ND) by the following transformation: {x,
t,
T, h,
~}—* {Xr~, trT~ Tr +
ATrl~hrh~~r~}
(D.5)
In this transformation, 6’ and Tr are the ND and reference (e.g. Hugoniot or release-state) temperatures of the material, while A T~is the difference between Tr and some “maximum” possible temperature such that 19 ~ 0(1). For example, if Tr is the release-state temperature, A Tr represents the difference between it and the reshocked-state temperature at the same pressure (see text). Substituting (D.5) into (D.2), we obtain 219 + —
~
V
~—
div h
= Dar~,.
(D.6)
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
397
In this expression, Per as X~itrKr is the Péclet number, Kr as kr/Prcpr is the thermal diffusivity, BOr as Pr~~Pr A Trihrtr is the Boltzmann number, and Da,. as ~ A Tr is the Damköhler number, of state r. Per, BOr and Dar are, respectively, the ratios of (1) free enthalpy flux to conductive flux, (2) free enthalpy flux to radiative flux, and (3) generalized internal energy supply rate to free enthalpy flux. We are particularly interested in Per and BOr~since their respective magnitudes will control the relative contributions of conduction and radiation to the energy transfer in the interior (away from the boundaries) of any layer of the target model. To obtain the magnitude of BOr~we first require an expression for hr, which may be obtained from (D.4). First, we assume ~e(A) is given by = jAb(nA,
A, T)
(D.7)
,
where tAb( A, T) is the Planck function and n5 is the index of refraction of the medium around p. Then, defining the Planck mean absorption coefficient (e.g. Siegel and Howell [15]), —
.f~° a5(A,
T, P)isb(nA, A, T)dA
(D.8)
1Ab~!~k’ It, ~ T\,1~ I )U/t
JO
and the incident mean absorption coefficient, f~a
(a)~~(T, P)
1A~dA 5(A,T,P)i~(A)dA as
coo
-
as
J
—
L 5(A) dA,
(D.9)
£5~~
and W/m2 noting K4that, if Stefan—Boltzmann n5 is independent constant, of A, j~° isb(nA, dA = in n OSBT , where 108 is the we may A, putT)(D.4) the form div h
41T[(a)~
as
0SB as 5.6696
x
2ff~~T4(a) —
1n
(D.10)
1~fl.
Nondimensionalizing (D. 10) with the appropriate transformations in (D.5) plus i—* i i~we have div h =
[(a)~~(1 +
where ~,as nro~suTr, hr as ~ ber, we obtain Bor
4lTartrnrtrsBTr pc AT =
r Pr2
r
~
~r6’)~
—
(a)111fl, and
(D.11) ~r
=
ATriTr~Substituting hr into the Boltzmann num-
(D.12)
From (D.6) and (D. 12), we see that radiative transfer will be an important means of energy transfer in an optically thick [Bo,.(a,.—*cc)--*0]and/or high-temperature medium, but not in an optically thin [Bor(ar—* 0)—* cc] medium, all other parameters being finite. At a boundary between two target components, we have a slightly different energy balance to consider. If we assume that the boundary is material and does not contribute to the balance of energy across it, the local balance of energy across the boundary between layers A and B takes the form
398
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
(D.13) where v is the “outward” normal unit vector to the interface. Using (D. 1) and the definition of hr given above, this takes on the ND form (VøA_~V19B).Vas~~~_ hA~v—~—
hBv,
(D.14)
B where ~ as kB ATB
S0
=
xA/kA
ATA XB, and Sk0 is the Stark number of layer 0, given by
k0AT0Ix0 2
(.)
4.
4ira0x0n0o~5T0
This parameter is a measure of conductive to radiative flux across a layer x0 thick with effective temperature gradient A T0/x0. Estimates of the parameters in these relations for the high-pressure states of the metallic DP, FL and dielectric SW are given in table D. 1. Of the parameters entering into Per, BOr~ and Skr~we note that Pr’ cpr, Tr~ATr, texp and ~1r are relatively well known (i.e. to within a factor of two) through the impedance match and equilibrium thermodynamic shock compression/release calculations discussed in the text. Most uncertain of all material parameters constituting Bo,. and Per, and so Skr as Bor/Per, are the thermal conductivity, k~, and the absorption coefficient, ar, of the high-pressure state of each layer. For perfect crystalline nonmetals k ~ lIT, while for metals k is constant at high temperature (e.g. Berman [59]; for minerals, see Roufosse and Klemens [60]), implying that the values of kr in table D. 1 are upper bounds if these dependences are relatively insensitive to pressure. If we assume that k for metals is dominated by its electronic component, ke~we can use the Wiedemann—Franz—Lorenz (WFL) relation [eq. (5.7) in the text] to estimate k from electrical resistance data, or some assumption on Pe’ since the relevant data are relatively scarce. Bridgman [70] investigated the change in electrical resistance of many statically compressed materials, including iron, but only at low pressure (s10 GPa). Keeler [39] investigated the change in the electrical resistance of shock-compressed copper and iron up to 140 GPa. He found that
Table D.1 Order of magnitude of parameters parameter
k,a)
T~ n~ a’~
metal
4
dielectric
iO~
i03 i0 102
101
i04 iol 1
iO~ iO~ 1
>106
10
SI units
kg/rn3
J/kgK W/mK
K K m~ rn
1o~ Pc, Bo,
<1 10
Sk,
10 ~ STP values.
lo~ <106
io~
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
399
Pe of shock-compressed Cu decreased from as16.7 to 5.6 nfl m with pressure up to —100 GPa; a datum
at 140 GPa implies that the resistivity of Cu reaches a minimum between 100 and 140 GPa and then increases to —0.83 nfl m at 140 GPa. As for Fe, the data imply that its electrical resistivity decreases from —2.5 to 0.47 nfl m between 13 and 140 GPa (above the a—* s transition). In light of (5.7), these trends imply that the thermal conductivities of shock-compressed Cu and Fe increase with shock pressure, against the high-temperature expectation expressed above. Consequently, we may assume with some basis that the STP value of k provides a lower bound to the high-P, T value of the metallic target components, k,.. The thermal conductivity of dielectric target components is dominated by the lattice contribution, k~, at high pressure; and we may estimate k from the lattice thermal-conductivity model of Roufosse and Klemens [60], who argue that the acoustic branches of the phonon spectrum dominate k~.Using this assumption, they arrive at an expression for k~,i.e., k~= (B’IT)[~(T1IT)112 + ~(T/T1)]
(D.16)
,
with T
‘l
=
213 _________ 1/6 25/3 2
n
(31T
)
Ma U 2 kBYP
‘
1
D’ ‘-‘
Ma
_________ —
2
1/6
3
2/3
ir
7/3
2
a n
1/3
2
where n is the number of atoms in the unit cell, Ma is the atomic mass, U, 1, is the velocity of sound, kB is 3 is the atomic volume, and is the acoustic-phonon Grüneisen parameter. Boltzmann’s constant, a This relation is consistent with the k ~ 1 / T expectation at low pressure. However, if the effect of pressure is to increase k, then k may change little from its STP value. Arguments in favor of this are given by, e.g., Roufosse and Jeanloz [71], who find that various two-body interatomic force models appropriate for halides predict an increase in k~with density and a decrease across polymorphic phase transitions with an increase in coordination. We use (D.16), (D.17), the solid-state equilibrium thermodynamic model presented in appendix B, and the equilibrium shock compression relations presented in the text to estimate high-P, T values of ke for metals and k~for dielectrics in our targets. These calculations, discussed in the text, are consistent with assuming the STP value of k for metals and nonmetals is a lower bound to the high-P, T values of k for these materials. As for the absorption coefficient, we expect the STP values given in table D. 1, like those for k, to be lower bounds, since absorption in the optical band is most likely to increase with pressure and temperature (e.g. Siegel and Howell [15]). Only a few initially transparent materials, such as A1 203 (Bass et al. [6]), retain their original transparency upon shock compression to high pressure (~250GPa). From the parameter values given in table D.1, we expect B0F ~ i0~m/aF and Bo~ 1010 m~/a~ for the FL and SW, respectively. Since we have no upper bounds on a~7and a~.,we cannot really say that radiation 7will important within the FLtransport and SW, should but it seems unlikely. What we can to saythe is m’never and bea~ ~ 1010 m’, radiative not contribute significantly that if a,. ~ i0 energy balance within the FL and SW, respectively, on the time scale of the experiments. Note that this bound for aF is probably underestimated, since we have assumed a rather large value for Tr in the DP or FL (table D.l). As for conduction, we have, from table D.1, PeF ~ 1012 m2 x~and Pe~~ 1012 m2 x~,and this implies that conduction will be significant over a length scale of s106 m in the DP, FL and/or SW. On
400
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
this basis, it is likely that conduction will contribute to the energy balance in both the FL and SW on the time scale of the experiment over a length scale equal to the thickness of the FL, since we expect k~, and especially kF, to increase with pressure and temperature (i.e. during shock compression). From all this, we believe that the values of BOr and Per given in table D. 1 are upper bounds to all relevant values of these parameters at higher pressures and temperatures. In order to emphasize the uncertainty of k~,a,. and the governing length scales, XF and x~,,of the FL and SW, respectively, we write (D.6) for the FL (or DP) and SW, and (D.13) for the FL—SW interface, using the values for the better constrained parameters from table D. 1, in the following forms (with A as FL and B as SW): —
tlr6’w
—
10’4(kF/x~)V219F+ 107a,. div hF
=
DaF~,.,
(D.18)
10~3(k~/4)V26’~ + 10~°a~ div h~= ~ WXF
k
.7
aFxF
3
V(9~~v+10j~—hFv10
FXW
F
(D.19)
xFxWaW
k
h~’v.
(D.20)
F
For a given layer to be, in effect, spatially infinite with respect to a given process (i.e., the boundaries have no effect on the process), the length scale should be, roughly speaking, about an order of magnitude less than the layer thickness. The conductive length scale is x,~ 0,,as ‘~/‘(rtexp,while that for radiation is Xrad as a’. The STP values of these length scales, given in table D.l, when compared to x,. of each layer, imply that both the DP and SW are infinite with respect to conduction, but only the DP is with respect to radiation. This implies that any conduction in the DP and SW, and radiative transfer in the DP, will be quite localized on the 1timel0~ scale thethis experiments. will shock pass essentially m of In case, only Radiation the interface, front and unhindered through the SW for ar ~ x USW free surface will significantly affect the radiation intensity. We consider the effect on radiation of propagation across the interface, SW, shock front, USW and its free surface in appendix F. By the same reasoning, we conclude that any conduction effects on energy transport affecting the observed radiation field will be confined to a region near the FL—SW interface on the order of the FL thickness. On this length scale, heterogeneous heating at the DP—FL interface and/or phase changes between the DP—FL and FL—SW interfaces may also contribute to the temperature field in the neighborhood of the FL—SW interface. In light of all this, we assume that XF and x~,are given by the corresponding values of ~ for the FL and SW, respectively. Putting these into (D. 18)—(D.20) and reducing them, we obtain ~.
—
—
102kF V219,. + 107aF div hF
=
DaF~,
(D.22)
10~k~ V219~= ~
V19F~v=(kw/ kF)V19F~v+ 105(a,./k,.)h,..
(D.21)
ii,
(D.23)
where we have eliminated the radiative terms for the SW in (D.22) and (D.23), since a~would have to be on the order of ~ m~before radiative transfer is important in the SW and across the FL—SW interface, regardless of the value of x~,.If a~is much larger (e.g. 108 m’) and kF much smaller (e.g. 10) than their values given in table D.1, radiative transfer could influence energy transport in the DP, FL, and across the FL—SW interface. However, for lack of better information, we adopt the values in
B. Svendsen
et a!., Optical radiation from shock-compressed materials and interfaces
401
table D. 1, thereby assuming that the thermal conductivity of the DP, FL and SW, and the absorption coefficient of the DP and FL, are effectively unaffected by compression/release. In this case, (D.21)—(D.23) become Per’ V2ØF
—
DaF~,
(D.24)
Pe~,’V2ø~,= ~
(D.25)
V19D.Vas~tV19W.V,
(D.26)
—
where {PeF, Pew) 1 (with XF x~) and js 10_i to 1. In this case, radiative transfer does not contribute substantially to energy transport in any part of the DP—TW system. So, we are left to investigate the effects of conduction at the interface through relations (D.24)—(D.26) in appendix E, and the effects of radiative transport through the FL—SW interface, SW, shock front, USW and its free surface by way of a model based on (D.3) in appendix F. ‘-—
Appendix E. Conductive transport model The scaling arguments presented in appendix D provide some idea of conditions under which conduction and/or radiation may be an important means of energy transport in the interior of each layer of the target, and across the boundaries between the layers. With some basis for believing that we are in the range of conditions where conduction dominates radiation as a means of energy transport, we decouple these processes and treat them separately. In this appendix, we establish a simple conduction model for the target geometry. We adopt the framework of Grover and Urtiew [9], who assumed that (1) significant conduction takes place only along the direction of shock propagation (i.e. is one dimensional), and (2) shock compression transfers energy to the material much faster than it can be conducted away. This last assumption allows us to treat the DP and SW as thermal half-spaces. Choosing the initial thickness of the FL, d, as the governing length scale, we have the ND energy balance for the compressed/released/reshocked DP (—cc < ~ < —1, assumed the same as FL), compressed/released/reshocked FL (—1
~6’A(~, r)= —PeADaA~’A(~, r) (r>0),
(E.1)
where A = DP, FL, or SW. We use the Laplace transform (LT) technique (e.g. Carslaw and Jaeger [58]) to solve (E.1) for each layer. Applying this transform to (E.1), we obtain V20A
—
[qA(s)]20A=
~
s),
(E.2)
where 0( ~, s) is the LT of ø( ~, T), s is the LT variable, q~(s)as \/~~ and 4’A(~,
s) as PeA[DaA~’A(~,s) +
19A(~,
which contains the ND initial conditions
0)], 19A(
~, 0). The general solution of (E. 1) is given by
(E.3)
402
B. Svendsen et a!., Optical radiationfrom shock-compressed materials and interfaces
0A(~,s)=bA(s)e~+ cA(s)e_~+ 0AP(~,s),
(E.4)
with
0AP(~,s)=~
f~e~_~—e_’~)~A(~,s)d~
being a particular solution of (E.2). The boundary conditions include the requirements that 19,.( ~ —cc and +cc, respectively. Consequently, we have and 19~(~, ‘r) remain bounded as ~
T)
~
0
cD
as—
“~
J
~)
‘t’D\b’ I
b
b~=
q~0 Using these and writing (E.4) for each layer, we have ~
s) as bD
~~Df+
s),
0Dp(~’
bFe~ + cFe
0,.(4,s)=
+
—cc< ~
(E.6)
0~
(E.7)
are now given by
~J J
0DP(~,s)=
0~(~, s)
=
(E.5)
0FP(~,s), —l~~
~ where 0,~ and ~
~ —l ,
~
e
~Je’~D(~,s)d~,
~D(~,s)d~+
e~’~~( ~, s) d~+ ~
[e~~w(~,
(E.8)
s) d~.
(E.9)
Relations (E.5)—(E.7) contain four unknowns, bD, bF and cF, and c~, requiring us to specify four more boundary conditions. We obtain these by assuming the usual continuity of energy flux and temperature across each interface in the target (appendix A.2). In the transform domain, these are given by —kD ATD
(l/s)TD
+
8f0~J(—l,s) = —kF ATF a~0, ATD OD(—l, s)
—kF ATF dfOF(O, s)
(1/s)TF
+
as
=
(l/s)TF
+
1(—1,s),
(E.10)
ATF 052(1, s),
(E.11)
—k~AT~c9f0w(O, s),
AT,, OF(0, s) as (1/s)T~+ AT~0~(0,s)
(E.12) .
(E.13)
B. Svendsen cc a!., Optical radiation from shock-compressed materials and interfaces
403
Putting (E.6)—(E.9) into (E.10)—(E.13), we obtain
II~DFe e~ 0
I
~ 1 1 —i ~
0
b
as ODFATD
~(
~~F 1 1
0 \ /ATD ATF bF bD\
I/b,\ b
2 O~WFI~ ATFcF J1b31’
(E.14)
—1/ \AT~c~/ \b4/
j- eD(~,s)d~, —1
r eD(~,s)d~,
(7’,.— TD)— ~
b2as
2q~
f
.1
a
0
b 3~~ as ATF
1
(e~’+e~F~)~(~, s) d~+ ~WFATW ~ 2q~ .1
s) d~, a
0
b4as~(T~_TF)—~-~ ~
2q~ .1
2q~0
Assuming the coefficient matrix is nonsingular, i.e., 1 + a~DF)(e e Li(s) = —(1 + ~WF)( where
)$0,
—
(qD~qF)
1/2
~WF
1/2 as (kDpDcPDIkFpFcPF)
(kWpWcPW/kFpFcPF)
~
IIDF_(O•DF1)/(tTDF+1),
the system (E. 14) has a unique solution. Noting that 1
Li(s)
=
(1
+ ~DF)(1 + ~WF)(1
—
1-~.JF~WF
e qD =
(1
+ ~DF)(1 + ~wF) ~
~
~WF)
~_~F(2m+t)
, for —1< ~y~ 52
~
1
we invert the coefficient matrix in (E. 14) to obtain e~(I
/ATD bD\ (ATFbF
I\ATW ATF ~ C~/ cF
=
a,, (l—rwF)
a,2 (oWF—l)rD,.
(1 + ~WF)UDF —(1 2
+ ~WF)~DF
2~DF
(1+UDF) (1
—
0DF)
2o~,,e~ \ /b,\ (1+~DF)rWFe~
e~’ (1
—
ODF)OWF ~
a 43
a44
I
b 3/I’
I \b4~ (E.15)
B.
404
1” + ~WFe_~)~
a11
a
43
Svendsen et aL, Optical radiation from shock-compressed materials and interfaces
_
(1
+
(1
+ oDF)(e
(TWF)(e
—
~WFe~)
—
p
~
,
e~F),
a,2
as
(1
a44
as
—(1
+
o-WF)(e’ + a.DF)(e~17+
e~F)
ii~
Expanding these and substituting them into (E.5)—(E.7), we obtain the solutions in the LT domain. We then transform these back to the time domain using the convolution theorem. For the calculations presented in figs. 5.2—5.5, we require the temperature field for —1~~ ~0 and 0 ~ and ‘r>O, i.e. in the FL and SW. For the FL, we obtain
J
—,
T(~,r)=TF+ATD
T
JY~(~, ~
~)d~d~+B(~,r)(TD— TF)
—a o
ff OT
+
ATF
G,,(~,~,r
d~d~+ E(~,T)(TF
~)
— ~)~F(~’
—
T~)
—1 0
a +
1
ATwJ~ j ~‘F~ ‘~,~,r
~
—
d~d~,
~)
(E.16)
00
~
~,T)
1DF / 1 \1/2 ~ ~DF IT peDT) 1+
as
—
B(~,r) as 1
G,,(~, ~,T)as
pw,, exp{—[(2m ‘~
~DF
~
+
+
a
~
m=O
1)
(v,.~,,~~,.)m[exp{—[(2m + 1)
—
—
1/
—
KFD(l + ~)]2w2}
KFD(l +
m
(~DF~wF)[erfc{[(2m
+
~DF Pn0
2
+~
1)
+
~]wF} v~erfc{[(2m —
+
1)— ~]wF}], (E.17)
i
1/2
~ITPeFT~
a
~
~
m=0
/
rexp{—[2m+(~—~)]2w~}— v~,,exp{—[2m—(~+fl]2w~} -yDFexp{-[2(m
+
1) +
(~+ ~)]2w2}
I +vDFvWFexp{[2(m+1)(~fl]wF} ~ x~ 1 exp{—[2m + (~~_ P~~exp{—[2m (~ + ~)]2~}
—
—
-vDFexp{-[2(m+l)+(~+~)]wF}
~~0,
+~FvWFexp{[2(m+1)(~~)]wF}, ~
11 ~,T)as ‘F~S~
1 +UWF ~WF /
1
\1/2
~
a
(vz~~°
m=0
x
[exp{—(2m~ + K,.wfl2w~} —
—
ii~
exp{—[2(m
+
1) +
4 + KFwfl2W~}],
B. E(~, r) as
1
Similarly, for 0
(7
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
erfc{[2(m +1) ~ (~F~wF)m[~F ~WF m0
+WF
~0,
+
~]w,,} erfc[(2m —
—
405
~)wF]].
(E.18)
i.e. the SW, we obtain
—1
T(~,T)=Tw+ATDf ~ 0 Or
+ATF J~ J —w ~ (~, ~ —1
~)d~d~+R(~,r)(T,,—
T~)
0
aT
+ATwffGw(~,~,T
~)d~ d~,
~
(E.19)
00
1 1/2 (irpeDr) ~
2TDF ~
~,r)as
(1 + (7DF)(1 +
~
x exp{—[(2m + 1) +
Q(~,r) as (1+ (7DF)(1 1
+ ~WF) ~
K,.w~— KFD(1 +
(~DF~wF)m erfc{[(2m
1/2
1”
/
1/2
~
—
~DFexp{—[2(m
+1) + ~ +
a
~ (~DF~wF)m
m=O
rexp{—[2m
I I
a
z.~ (~DF~WF)
—
1 / 1 ~ ~irPe~r/
1) +
maO
x [exp{—(2m ~+ ~,r)
+
‘~
—w” ~ ‘~,~,r)as~)
~
~
+
KFw(4~—~)]2w~) + ~WFexp{—[2m
~,.exp{—2(m
+
1)
+ K,.w(~+
+ KFW(~+ ~)]22}
—~FvWFexp{—[2(m+1)+KFW(~—~)]wF},
x~ I exp{—[2m ~DF
+ K,.w(~—
exp{—[2(m
+
~)]2w~}+ ~WFexp{—[2m 1) +
+ KFw(~+
~)I2w2}
KFW(~+
—vDFpWFexp{—[2(m+1)+KFW(~—~)]2w~}, 4~~cc,
R(e, T) as 1+1
~
(VDFPWF)m[erfc{(2m
~WF m0
Each of the source terms now has the form
+ KFw~)WF) +
~,.
erfc{[2(m
+1) + KFw~]W~)].
406
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
PeA[DaA~’A(~
~A( ~, T) as
r) + 19( ~, 0)~(r)],
(E.20)
where 6(i-) is the Dirac delta function. As discussed in the text, we assume that compression/release processes in the target establish the initial (t as 0) temperature profile (5.2) in the DP—FL—SW system. Assuming ~D( ~, 0) = 0, ~( ~, 0) = 0 and ~‘F(~’°)—O’we have ~w(~,T)O, 4D(~’‘r)
=
PeD[h(~ + 1 + 6~) h(~+ 1)] ATDF/ATD, AT —
AT
(E.21)
~ ATF +PeF[h(~+~FW) h(fl] ATF from (5.2) and (E.20) written for the DP and FL, respectively, where h(x) is the unit-step function, and
6~’as61/d.Putting (E.21) into (E.16), we obtain the temperature in the FL (—1~s0, r>0), ~ +
E(~,T)(TF
—
(E.22)
T~), m[erfc{[(2m
A(~,T)as 1 —
DF
m=O ~
+
1)+ ~]w,,) —erfc{[(2m
+
1)+
~ + KFD8~]wF}
(VDFVWF)
v~,,(erfc{[(2m + 1)— ~]w,,}—erfc{[(2m+ 1)— ~ + KFD3~,]WF})],
C(~,r)asg~(~,r, —1, ~)+g~(~,T,
—l+6~),
4,
~
(E.24)
~1+ôFD,
(E.25)
~
~ g~, r, a, b) as
~
(~~FvWF)m[±erfc{[2m ± (a
+
vwF(erfc{[2m
—
PDF(erfc{[2(m
i
—
+
(~ +
a)]w,,}
—
—
~)]w,.} ~
erfc{[2m
—
erfc{[2m ±(b
(~ +
+
1) ~ (a
—
~)]w,,} erfc{[2(m —
—
b)]w~})
1) +(~+ a)]w,,} —erfc{[2(m+ 1)
VDFVWF(erfc{[2(m
(E.23)
+ (~+
+
1)
~
b)]WF})
(b
—
(E.26) The function g~( ~, T, a, b) results from the integration of the second term on the right-hand side of (E. 16) using (E .21); a and b are the lower and upper limits of the spatial integration, respectively. We obtain g~from g~by using the upper signs in each term having both + and minus signs, while g~is given by the lower signs in each term. Lastly, with 4w( ~, r) as 0 in (E. 19), we obtain the temperature in the SW (0~4, r>0), —
B.
407
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
T(~,r) Tw+A(~,r)ATDF+Q(~,r)(TD—TF)+Q(~,r,—1,—1+6D)ATFD +
fi(~,r, —o~,0)AT,,~ + R(~,‘r)(TF ~DF
—
Il
\I1
x [erfc{[(2m r, a, b)
V m \ ~ (~DF~WF) °WFI m=O
+
1)
+ K,,w~+
KFD(~l~]WF}
m[erfc{(2m 1~WF m—0 ~ (t/DFPWF)
1
as
L
(E.27)
T~),
+ VDF(erfc{[2(m +
1)
+ KFW~+
+ erfc{[(2m +
+
1)
KFw~ b)WF}
b]w,,}
—
—
erfc{[2(m
+
—
erfc{(2m
+
1)
+
K,,w~ a)wF} —
+ K,,w~+ a]WF})].
Relations (E.22) and (E.27) are those used in the text to examine the effect of conduction on the temperature field around the FL—SW interface.
Appendix F. Radiative transport model As discussed briefly in the text, radiation from sources at the FL—SW interface and/or the SW may be affected by radiative transfer through the SW, shock front, unshocked TW and its free surface. In general, radiation sources include (1) the FL at the interface, (2) gas or other trapped material at the interface and/or (3) the SW. The FL is generally a metal (i.e. an opaque material), while the TW is generally an oxide, halide or silicate. Many of these latter materials are of natural origin, possessing color and containing inclusions of various sizes, some of which are potential scatterers (e.g. apparent Rayleigh scattering in CaO: Boslough et al. [72]). These possibilities motivate us to formulate a model from (D.3) and the geometry of the target of sufficient generality to deal with emission, absorption and! or isotropic scattering in the shocked and/or unshocked TW. Note that in writing (D.3), we have already assumed isotropic scattering. For tractability, we must accept this as a limitation of the model. We have no a priori reason, of course, to expect this to be true in shocked or unshocked TW material containing scatterers. Since our experiments are calibrated, we need worry only about the differences between calibration and experimental configurations, which include radiation sources, and the effects of the (1) SW, (2) shock front, (3) USW, and (4) TW free surface, on the source radiation. For simplicity, we assume the target may be approximated optically as a set of parallel plane layers. In this representation, the temperature and material properties of the FL, SW and USW are assumed to vary at most only along the direction of shock propagation, x. Further, we assume that i~is axisymmetric about s; in this case, i~ depends only on s = s~and the angle between s and x, which we designate 4, so that i~= i~(s,~5). Introducing the extinction coefficient, K 5 as a5 + (75, the differential opacity, d K5 as K5 (s) ds, and the albedo for scattering, ~ as o-5/K5, into (D.3), with ds as dx/cos 4, the radiation intensity in all directions forward (‘—b +) and backward (‘—s. —) in a plane layer is given by the solution of (e.g. Siegel and Howell [15]) /_).+I
\\
( at5 ~K5, /L, I ) ilK5
.+
~ /2
.,
I-)
t~
~so
I
.+
[15(K5,s
*
.—
)+z5(K5,—~
*
)]dp.
*
,
L 0
(F.1)
408
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
) aK5
+
z,,, (K5,
—p~)=
(1—
+
In these relations,
(Iso) lse( K5,
—~)
/2
~
f
[i~5,
~*)
+
i~(K5,_~*)] d~*.
(F.2)
j~as C05
(F.3)
Ks(x)asJK5(~)d~
is now a function of x. Radiative transport in each layer of the target is then governed by (F. 1) and (F.2). Given a target geometry like that of figs. 2.1 and 2.2, we assume that observed radiation is dominated by radiation from the target of nearly normal incidence (i.e., ~ as 1). This is the fundamental simplifying assumption of the radiative transport model used in our work. Mathematically this assumption takes the form i~(K5,~
as i~(K5)ô(1 —
p~,
(F.4)
where 5(x) is Dirac’s delta function. In effect, this assumption imposes a very special averaging on the optical properties of the model, i.e., that model properties are dominated by incident radiation perpendicular to the layering of the target. Putting (F.4) into (F.1) and (F.2), we obtain di~/dK5+ i~=
(1
—
—di~1dK5+ i~= (1
—
+
as
~Q50(i~ + i~), +
~so)~e(”s)
respectively. Now let ~ as subtracting them, we obtain
dI5JdK5
+
~so)~e(’~s)
±~
(F.5)
(F.6)
~Q50(i~ + i~),
Substituting 1~÷ and I~ into (F.5) and (F.6), adding and
2I3~~,~e(Ks) + ~so’s÷ ‘
(F.7)
d15~/dK5+15_=0, with
\/i
$~=
—
(F.8)
-
Q~.Eliminating I~ from these expressions gives us a differential equation for I~ i.e., +~
21 d
5÷/dK~f3~I~= ~2/3~~,~e(Ks).
(F.9)
—
For ~ ~ some f(K5), which is the case if we assume that a5Ia-5 ~f(K5) general solution of (F.9) is given by
I5~(K5)= C e~
+
D e~
+
J
in each layer of the model, the
sinh{~5(K~ K5)) ~~e(K~) dK~. —
(F.10)
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
409
Then (F.8) provides
J
1 5(K5) =
—d15~!dK5as —~5Ce~
+
~5D e~
+
~
cosh{~5(K~ K5)} —
1~e(K~) dK~.
(F.ll)
Combining (F.10), (F.11) and the definitions of 1~±, gives us i~(K5)= ~
± J~)=
J
i~(K5)= ~
~(1 ~ ~5)Ce~
[(1 ±$~) ~
~(1 ± f35)D ~
+
—(1 ~
+
$~)e~]i~~(K~)
(F.12)
i~(K5),
dK~.
From (F. 12), we have the following conditions at the boundaries of the layer for backward and forward propagating radiation: ~(1+/35)C+
i(0)=
i~(KSd)= ~(l
+
~(1—f35)D,
— /35)D
~
+
$5)Ce~”~’~ + ~(1 + f35)D
~
+ i~P(KAd)
f35)C e~”~ + ~(l
i(0)= ~(1—f35)C+ i~(KSd)= ~(1
—
~(1+/35)D
where K5,~= K5(d), with d the layer thickness. We need only two of these to find C and D; for reasons apparent later, we choose the second and third relations. Solving these for C and D, and substituting the resulting expressions into the first and fourth relations, we obtain
(
i~ 0
—
2(1
4PS[i~(K e~2d 5d) i~P(Ksd)] ,
13~)(1 sinh(f3SKSd) + e~2d i~(0)(1+ —
—
~)2
—
~)2
4f35i,~(0)+ 2(1 f3~)sinh(I3SKAd) [i~(KAd) i,~P(KAd)] (1 + ~)2 e/~K20j—(1 e~~’~’°” + 1SP(KAd) —
t~(,ç~) =
( F . 13 )
—
—
—
~)2
(F.14)
.
—
With f3~=
~ and KSd = Kss,., these relations apply to the SW; with f3~= $susw’ Ksd = K5FS K5SF and 0 (no sources), they apply to the unshocked TW. Next, we assume that (1) the opacity of the FL is sufficiently large so that any radiation observed from it originates near the interface, and (2) all boundaries are optically smooth, so that radiation incident on these boundaries is refracted and reflected according to Fresnel’s laws (e.g. Siegel and Howell [15]). Basically, by doing this, we neglect any scattering properties these boundaries may have. Using (F.13) and (F.14), and assigning each boundary of the target an intrinsic normal spectral reflectivity, r5, given by (Siegel and Howell [15])
i~P(KSd)=
2 + (w5A rAAB as [(~5A
—
~AB)
—
~5B)]’[(~5A
+ ~5B)
+ (w5A +
WAB)]
410
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
for normal incidence on the boundary between layers A and B possessing complex indices of refraction t05A and ~5B + la/SB (where w ~sA + ‘ 5 ~ a5), respectively, we may construct the boundary and layer conditions for the model. At the interface (x = K5 = 0), radiation traveling forward into the SW, i ~~(0)~ is composed of radiation emerging at the SW side of the interface from the FL and! or gas, S ~(0), plus that component of the backward traveling radiation, i ~~(O), reflected off the interface back into the forward direction, i.e., i~~~(0) = S~1(0)+ r5ii~~~(0).
(F.15)
This is the boundary condition at the interface. From (F.13) and (F.14), we have conditions for backward and forward propagating radiation in the SW [0~ x ~ xx,,, 0< K5 < KSSF = KS(xsF)], i.e., ~sw(0)
=
c1i~~~(0) + c2i~w(K5s,,)+ S~w(KssF), c2i~~(0) + c1i~~(K55,,) + S~sw(KssF),
j~sw(KssF)=
respectively.
(F.16)
At the shock front [x = XSF, KSSF
=
(F.17)
KA(xsF)], radiation is transmitted and reflected in both
directions. Consequently,
(F.18)
i~sw(K5s,.) = c3IAUSW(KSSF) + T5SFIASW(KSSF) ,
~~usw(’AsF) = c4iAsw(KAsF) + rASFLAUSW(KSSF)
(F.19)
.
Again using (F.13) and (F.14), the conditions in the unshocked window KAFS = KA(XFS)] are given by
[XSF
~ x ~ XFS, KSSF ~ K5 ~
iusw(K5s,,) = c,iUsw(KASF) + c6i~~Usw(KAFs) ,
(F.20)
j~~~k552~ ) = c6iusw(KAsF) + cSZAUSW(kAFS) ,
(F. 21)
where we have assumed ~e~~(K~) as 0, i.e. no emission from the unshocked window. Lastly, at the unshocked window free surface [x = XFS, KSFS as KS(xFs)], radiation traveling back into the unshocked window, iusw(KSFs)’ is equal to that component of the forward traveling radiation incident at the free surface, i~usw(Ks,,s) [x = XFS, KSFS = K5(x,,5)], reflected into the window, i.e., i~USW(KAFS)= rAFSZAUSW(KAFS)
(F.22)
.
The coefficients and source terms in these relations are given by
—
c,= 2(1
—
~
Asinh(J3SSWKASF)
4I3~sw
fl
,
3—i Asw
c
=
5—
c
,
2(1
—~
Ssw 413S~w
sinh[f35Usw(KSFS
—
K5SF)]
=
, AUsw
c6_,~ 5Usw
rSsFasc4,
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
S
+
—
.+
(1
51(0)
—
rsj)lsei(O)
35sw
/
—
A
~ssw(1(ssF) as
‘
411
l5sWP(KASF)
55W
2(1
.÷
÷
S5s~(K5s52)as 1SSWP(KSSF)
—
~
sinh(/355~K5552)
A
—
5
lSsWP(KASF)
SW
where (1 + Asusw
=
/3)2
ePAsw”¾5b2—
~l + t-’susw) ~
~2
‘.
(1
—
e~1\~55F
$)2
~
—
\
—
eu1F~~~sb2)
~ ~ t-’sUSWI
Relations (F.15)—(F.22) constitute 8 equations in 8 unknowns. We are particularly interested in calculating i~usw(K552s),since the radiation escaping the target destined for detection is then given by tsmod(KSFS) as
(1
— rSFS)IAUSW(KSFS)
(F.23)
.
The system (F.15)—(F.22) may be cast into the following matrix form: S~~(0)
—r51 1 0 0 0 0 0 0
0
0
0
0
0
—c1—c2
1
0
0
0
0
0
0
_~2
~
_~,
0 0 0 0 0
1 0 0 0 0
i~5~(0) .-
~
rSSF
c4
c3
rASF
0 0 0
1 0 0
+
155w(K5SF)
0 1 —c, c6 0
0 0 —c6 —c5 1
0 0 0 1
S~5~(K5552) S5SW(K5SF)
1ASW@~ASF)
i~ (K
~
0 0
5SF
t5usw(KASF) i ~‘K~
rFS
0 0
SUSW\ SFSI
i~usw(K552s)
0 (F.24)
In matrix notation, this is [A]{i} = {d}, where [A] is a reflectivity—transmissivity matrix, {i} is a vector of the forward and backward traveling radiation intensities at the boundaries, and { d } is a radiation source vector. Decomposing [A] into upper and lower bidiagonal matrices, [A ~] and [AL], respectively, we find the solution of [AL]{u) = {d} and use this to solve {i} as [A~]’{v}. Solving this system gives us an expression for iU5W(KAF5), which we substitute into (F.22) to obtain jSfflOd(KSFS)
as
~P~[S51(0)+ SSSW(KSSF)]
(F.25)
.
In this relation, ~ is the effective transmissivity of the SW, shock front, unshocked TW and its free surface, given by =
E~(1 r55F)(l
as
[(1
—
— r51c1)(l
—[(1
—
—
—
r5,,5)c6 ,
c1r5552)
— r51c~r5~,,][(1
—
r51c1)c1 + r51c~](1
—
r5SF)
[(1
—
rASFc5)(l
c5rAFS)c5
—
cSrSFS)
+ c6rSFS]
—
rSSFc6rSFS]
412
~
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces is
the intensity of sources in the SW along K5, i.e., A SF
f
SASW(KSSF)as
A5(K55~, ~
0
1 As(KssF,
~) as
~
3~~~) [c, + (c~
r
—
/3ssW{(
—
51c1)(1
+
— ~/3~~~{(1— r51c,)(1
—
I —
c~)r 51](1— ~3~~)} ew~F)
$~~) —
[c1 + (c~ c~)r51](1+ /35sW)} e_w_K25~, —
and S51(0) is the combined effective intensity of all interface sources, S51(0)
c2(1
as
— rsj)i~ei(0)
Relation (F.25) is the principal result of this appendix, and constitutes the simplest model incorporating scattering, absorption and multiple reflections that we can derive for radiative transport in the target. As it stands, (F.25) is sufficiently general for comparison with observations from a number of different radiation experiments. At this point, however, our main interest is in the interface experiments discussed in the text, so we now specialize (F.25) to this purpose. For cases3SUSW where= there is apparently scattering in the SW, and none in the unshocked TW, we may 1 in (F.25); in thisno case, c set ~ = / 1 = c5 = 0, c2
as =
c6
as e’~5F
(1
— rSFSI-AUSWrASF)
AA(KSSF,
~) = rASW(e~+
as ~
=
—
TASW[rSFS(l
(F.26)
,
—
2r55,,)c~+
(F.27)
rSSF]rAI ,
(F.28)
r51 e~) .
From (F.27), we note that, in general, neglect of multiple reflections when {r5, r5}—* 1 is clearly incorrect and can lead to underestimate of the source intensity by the model. However, for the experiments of interest here, we do have r5 0. In this case, 2rASF)rSI = 1 0(r2), (F.29) H5(A, t) ~ 1 — (r5FS + rAI)rSSF — rAFs(1 and multiple reflections are, to first order, negligible. Now if we assume each region of the dynamic target optical geometry possesses a distinct extinction coefficient, and that, in each of these regions, —
—
—
optical properties are homogeneous, we have
K 55~, K5(x)= KSSF, ~
O
(F.30)
XSF
where x~,,is the position of the shock front in the SW, x~Fits position in the USW, and
4,,
—~
is
the
shock front thickness. Putting (F.30) into (F.3), we obtain K55,,(X) = KASWX~F,
(F.31)
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces KAFS(X) = K
55,,(x) +
K5~,,(x~52xsF) —
+
413
(F.32)
K5~5~(dx~F). —
For steady shock wave propagation, x and t are not independent, and we may write x~,,(t)= (u
—
x~F(t)= (u
U)t,
u~)t,
—
(F.33)
where u is the speed of displacement of the shock front, and u - and u are the material velocities behind and in front of the shock front (notation of appendix A.1). Putting these into (F.31) and (F.32), we have +
KSSF(t)
=
KSSW(u
KAFS(t)
as KASF(t)
—
(F.34)
u)t,
+ K5sF(U
—
v~)t+ K5~~~[d(u —
—
u~)t].
(F.35)
Defining as
~
K55~(u
—
= KSSF(U as
—
U)texp,
(F.36)
U~)texp‘
(F.37) (F.38)
~
as nondimensional extinction coefficients, and setting u —
j,’*
as
0, (F.34) and (F.35) may be written
~
(17
3
ASW ‘‘exp ,
—
KSFS(t)
+
KS5F(t) + K~SFt/texp+
K~’~~~(1 t/texp) ,
(F.40)
—
respectively. If we treat the shock front as a 2D boundary, rather than a thin layer (appendix A. 1) K55,,—~O.In this case, (F.40) simplifies to KSFS(t) = K5s,,(t) +
K~’UsW(1
—
t/texp)
(F.41)
.
Relations (F.39) and (F.41) are the expression we use in (F.26) and the text. Lastly, if we assume that sources in the SW are distributed uniformly, ~ ~) is independent of x, i.e. spatially uniform, and from (F.22) this implies KASF
SASW(KSSF)
assuming that
as
f
i.~eSWis
AA(KSSF,
~)~~esw(~)
d~as (1
—
r5~~)(l + rSIr5SW)IA+(A, T~),
(F.42)
given by the Planck function. Putting these results into (F.25) and (F.26), we
obtain, with S~= 1, tAmod(A, t) as êssW(t)ISpl(A,
T~)+ â51(t)I5~1(A,T1(t))
,
(F.43)
414
B. Svendsen ci al., Optical radiation from shock-compressed materials and interfaces
where
e5~~(t) = ~1~(1
—
TASW)(l +
r51’r55~),
ê51(t) as ~P~TAsW(l
—
r51)
are the effective normal spectral emissivities of the SW and FL at the FL—SW interface, and as (1 — rAAB)T5USW(l — r5SF) with E,~= 1. Relation (F.43) is the expression we use in the text, with tAmod~
The Hugoniot temperature of the SW, T~,is assumed homogeneous, uniform, and
constant with a uniform distribution of SW sources. The interface temperature, T,(t), however, may be ‘Amod a function of time, or constant, in the context of the conduction model discussed in appendix E.
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416
B. Svendsen ef a!., Optical radiation from shock-compressed materials and interfaces
168] FE. Prieto and C. Renero, Equation for the shock adiabat, J. AppI. Phys. 41(1970)3876—3883. [69] D.J. Pastine and D. Piacesi, The existence and implications of curvature in the relation between shock and particle velocities for metals, J. Phys. Chem. Solids 27 (1966)1783—1792. 2, Proc. Am. Acad. Arts Sci. 81(1952)167—251. [70] TheR.resistance 72 elements, alloys of and compounds 100,000kg/cm [71] P.W. MC. Bridgman, Roufosse and Jeanloz, of Thermal conductivity minerals at highto pressures: the effect of phase transitions, J. Geophys. Res. 88 (1983) 7399—7408. [72] MB. Boslough, T.J. Ahrens and AC. Mitchell, Shock temperatures in CaO, J. Geophys. Res. 89 (1984) 7845—7851.
Bob SVENDSEN Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA Jay D. BASS Department of Geology, University of Illinois, Urbana, IL 61801, USA and Thomas J. AHRENS Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
I
NORTH-HOLLAND AMSTERDAM -
PHYSICS REPORTS (Review Section of Physics Letters) 180, No. 6 (1989) 333—416. North-Holland, Amsterdam
OPTICAL RADIATION FROM SHOCK-COMPRESSED MATERIALS AND INTERFACES Bob svENDsENa)*, Jay D. BASSb) and Thomas J. AHRENsa) Seismological Laboratory, California institute of Technology, Pasadena, CA 91125, USA ~ Department of Geology, University of Illinois, Urbana, JL 61801, USA Received April 1989
Contents: 1. 2. 3. 4.
Introduction Model considerations Model assumptions Initial conditions for conduction and radiation 4.1. Shock-compressed state 4.2. Release and reshock states 4.3. Application to Fe targets 5. Conductive transport in the target 6. Radiative transport in the target 7. Models and data
335 335 338 339 340 350 354 357 365 367
8. Summary Appendices A. Shock front and target boundary conditions B. Equilibrium thermodynamics and shock compression C. Isentropic release and reshock D. Energy transport in the target E. Conductive transport model F. Radiative transport model References
375 377 377 387 391 395 401 407 414
Abstract: Recent observations of shock-induced radiation from oxides, silicates and metals of geophysical interest constrain the shock-compressed temperature of these materials. In these experiments, a projectile impacts a target consisting of a metal driver plate, a metal film or foil layer, and a transparent window. We investigate the relationship between the temperature inferred from the observed radiation, and the temperatures of different high-pressure states (e.g. shocked state) of the shock-compressed film (or foil) and window. Deviations of the temperature in each target component away from that of their respective shock-compressed states occur because of (1) shock-impedance mismatch between target components, (2) thermal mismatch between target components, (3) surface roughness at target interfaces, and (4) conduction within and between target components. In particular, conduction may affect the temperature of the film/foil material at the film/foil—window interface, a major thermal radiation source in the target, on the time scale of the experiments. To be observed, radiation from sources at the film/foil—window interface or in the shocked window material must propagate through (1) the shocked window material, (2) the shock front, (3) the unshocked window material, and (4) the unshocked window free surface. Consequently, the observed intensity of target radiation sources is affected by the optical properties of each region. In particular, the source radiation intensity may be greatly reduced due to absorption in the shocked and/or unshocked window material, and/or only partial transmission through the film/foil— window interface, shock front and/or unshocked window free surface. To illustrate various aspects of the model, we apply it to radiation data from targets composed of an Fe driver plate, and Fe film or foil layer, and either an Al 203 or LiF window layer. * Present address: Department of Mechanics, Darmstadt Institute of Technology, D-6100 Darmstadt, FRG.
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B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
335
1. Introduction Traditional studies of the behavior of shock-compressed materials assess the mechanical response of these materials to shock compression (e.g. the change of density with pressure). Since this approach cannot directly constrain the temperature of the high-pressure state, other means are needed to provide experimental constraints on a complete equilibrium thermodynamic description (i.e. pressure—density-temperature) of these materials. To this end, recent experimental efforts observe and record shockinduced radiation; both initially transparent materials (e.g. alkali halides, summarized by Kormer [1], A1203: Urtiew [2], Si02 and Mg2SiO4: Lyzenga and Ahrens [3]) and (2) opaque materials at interfaces viewed through transparent or semitransparent windows (e.g. Mg: Urtiew and Grover [4], Ag: Lyzenga [5], Fe: Bass et al. [6]) have been investigated. This radiation, at least at high pressure, appears to be dominantly thermal, and so constrains some temperature in the target. In this paper we explore possible relationships between the experimentally constrained temperature and the temperatures of different high-pressure states achieved in the target components and at their interfaces during the experiment. We attempt this in the context of a simple model of energy transfer and transport in the targets, first dealing with conductive transport, and then radiative transport, which may be considered separately for the experimental conditions and materials of interest. We begin by discussing how shock compression, release, reshock and phase transitions may influence the temperature profile in the target (section 2), arguing in particular that these processes operate on a much shorter time scale than energy transport by conduction and/or radiation (section 3). We then discuss the likely importance of radiation versus conduction (section 3, appendix D) to the target energy balance via dimensional analysis, which implies that conduction will likely dominate radiation as a means of energy transport at high pressure in the materials of interest here. After these general considerations, we begin the detailed development of (1) the initial conditions for energy transport in the target as established by shock compression, release, possible reshock and phase transitions (section 4, appendices A, B, and C), and separate (2) conduction (section 5, appendix E) and (3) radiation (section 6, appendix F) models based on idealizations of the target geometry. Finally, we discuss the possible use of the combined conduction—radiation model, and certain simplifications of it, to fit data (section 7).
2. Model considerations Consider the target depicted in fig. 2.1, representative of that used in the experiments of Lyzenga [5] and Bass et a!. [6]. This generic target consists of (1) a 1.5mm thick metallic “driver” plate (DP), (2) a metallic film (1 to 10 ~imthick) or foil (10 to 100 ~imthick) layer (FL), and (3) a dielectric, transparent window (TW, 3 to 4mm thick). The target is constructed such that the shock impedance (i.e. the product of the starting density and shock wave velocity) of the DP is greater than or equal to that of the FL, which in turn has a shock impedance greater than or equal to that of the TW. An edge mask (fig. 2.1) prevents the detectors from sensing radiation from the edge of the target assembly, where conditions are probably inconsistent with the model for shock compression used in this work (see appendix A). Radiation emitted from the center of the assembly, where uniaxial compression most likely occurs, reflects from the mirror into the detectors. The experiment begins when a projectile impacts the DP (fig. 2.1), generating a shock wave that propagates through the DP to the DP—FL interface. Since this interface is formed by mechanical
336
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Pyrometer
Driver PI~1~
~Mirror
~
Driver Plate
Film Driver Material
-
Window
Material
Interface
Foil or Film
Fig. 2.1. Target assembly.
juxtaposition of the metallic DP and FL surfaces, it is “rough” on a —1 ~m scale (Urtiew and Grover [7]). The shock front thickness is ~0.01 p~min the materials and at the pressures of interest (e.g. Kormer [1]). With respect to the shock wave, then, the DP and FL surfaces are, prior to compression, partially free. Consequently, the shock wave accelerates the DP material at the DP—FL interface (which compresses the DP—FL interface) and simultaneously reflects from the DP surface at the DP—FL interface as a release wave propagating back into the DP, which releases the DP to near-zero pressure. The moving DP surface impacts the FL surface, generating shock waves of approximately equal magnitude that propagate backward into the just released DP, and forward into the unshocked FL. The former shock wave compresses the just released DP material from its low-pressure, high-temperature release state to one with approximately the same pressure as its previous compressed state; wave reverberations quickly bring this DP state to a state with a pressure equal to that of the compressed FL and a temperature well above that of the previous (i.e. first) DP compressed state. If the backward propagating shock wave overtakes the release wave at some distance behind the DP—FL interface, this distance defines the thickness of a reshocked DP material layer at the DP—FL interface. However, if the release wave is faster than the reshock wave, the entire DP may experience low-pressure release and reshock. In either case, subsequent wave reverberations quickly bring the DP to a state with the same normal (to the interface) material velocity and stress fields as the shocked FL. Since the DP material accelerating across the DP—FL interface impacts a rough FL surface, a thin (on the scale of the surface roughness) layer of film or foil material compresses, much like a porous material (Urtiew and Grover [7]), to a much higher temperature than achieved by the compressed FL material beyond this zone. As with the DP material at the DP—FL interface, the shock front traversing the FL reflects from a partially free surface at the FL—TW interface as a low-pressure release wave and accelerates the FL material across the interface to impact with the TW material. Since the TW surface at this interface is smooth relative to the shock front thickness, and the TW is much, more incompressible than either the DP or FL, the impacting FL material should not heat a thin layer of TW material, but rather only shock the TW up to high pressure and its Hugoniot temperature. The encounter of the shock wave propagating through the FL with the FL—TW interface generates
B. SventLsen et al., Optical radiation from shock-compressed materials and interfaces
337
backward and forward traveling shock waves, and the former wave compresses the low-pressure, high-temperature, released FL material to a state with approximately the same pressure as the first FL compressed state; wave reverberations quickly bring this FL state to a state with a pressure equal to that of the compressed transparent window (shocked window: SW) and a temperature much higher than the first FL compressed state. If the backward propagating shock wave overtakes the release wave, it cuts off the zone of release / reshock in the FL material. In this case, the combined wave releases the remaining FL material, and then the DP, to a state with approximately the same normal velocity and stress as the SW. Alternatively, if the shock wave does not overtake the low-pressure release wave, the entire FL and/or DP is released and reshocked. In either case, subsequent wave reverberations should quickly bring both the DP and FL to states possessing normal stress and material velocity fields equal to those of the SW. As the FL material at the FL—TW interface is compressed, released, and possibly reshocked, it heats up and begins to radiate. Consequently, the observed radiation intensity rises sharply (fig. 2.2A,C). As the shock wave travels forward into the TW, the thickness of the SW increases (fig. 2.2B); consequently, so does its contribution to the total observed radiation (note increase with time in fig. 2.2C). If the SW is highly absorbing and/or scattering (apparent in many experiments, Boslough [8]), the radiation intensity from the interface will decay with time (fig. 2.2C, dash—dot curve labeled fast decay); if not, the interface source will dominate the observed radiation history (fig. 2.2C, continuous curve labeled slow decay) since the FL material at the FL—SW interface is almost certainly at a higher temperature —Shocked Film — Partially-Released —
i—
Driver ‘Plate
~
1
(A)
Release Front
/
SW
FL
/~
x
“‘e
I NT ~
TOTAL foot decoy
~
~
decoy
I dec
0
~
I
slow decoy
Enters Target” ~
~
+
~sIow
Shock Wave //
~
/iSW
Shock Front
rs
I
DP
I
‘INT
Shock:d —Window ~a~riol
~
~:~j-~
Film
Film-Shocked Window Material Interface
i~~fl
~
Shock Wave Reaches Interface t — t INT
(C)
TW (B)
=
Fig. 2.2. Dynamic model geometry. Shock front reaches film/foil layer (FL)—transparent window (TW) interface (x 0) at time t1, when radiation is first detected. Interface radiation (lINT) dominates the early radiation history (A). If (1) the interface temperature decays slowly, and (2) the FL—TW interface, shocked window and shock front remain relatively transparent, LINT will dominate the observed radiation history during the experiment (“~5~ + slow decay” curve). However, if the FL—1’W interface and/or shock front develop significant reflectivity, and/or the SW develops significant opacity, ~INT will decay quickly (dash—dot curve), and may fall below the radiation intensity of the SW, ~ on the time scale of the experiment. The total intensity is then represented by the “i~~ + fast decay” curve.
338
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
than the SW. Assuming the observed radiation represents the sum of the interface and SW contributions, the interface contribution will dominate the observed radiation intensity unless the SW is strongly absorptive. Since the reshocked layers at each interface are significantly hotter than the surrounding material (see Urtiew and Grover [7], and discussion below), the temperature and radiation histories of targets with smooth versus rough interfaces should be quite different, and perhaps even experimentally resolvable. We investigate the dependence of the radiation history on the nature of the interface by examining radiation data from both mechanically formed foil—TW interfaces and vacuum-coated film—TW interfaces. In particular, we expect the vacuum-coated film—TW interface to be ,much smoother than the mechanically formed foil—TW interface. This expectation turns out to be somewhat naive, however, as shown below. Since the TW surface at the FL—TW interface is smooth (as defined above), we presume that any roughness of this interface is due to roughness of the FL surface. Besides conduction, dynamic phase transitions or other energy sources and/or sinks present in the FL, FL at the interface, and/or SW on the time scale of the experiment may also introduce time dependence into the temperature and effective emissivity inferred from the radiation observations (Grover and Urtiew [9]). In this study, we limit ourselves to the processes of conductive and radiative transport. With respect to the latter process, we must account for the effects of propagation through the SW, shock front, unshocked window (USW), and the TW free surface on the FL—SW interface and SW source radiation intensities.
3. Model assumptions We assume all sources contributing to the observed radiation intensity are thermal and in local thermodynamic equilibrium. We can then relate the source intensity to the wavelength A, and absolute temperature T, through the Planck function IApl(A, T), given by I5~1(A,T) where C
5(eC2~T—
1)’
(3.1)
A
7Wm2/sr and C 1 ~5.9544 X i0~” 2 1.4388 X 10’~mK are constants. Comparison of the observed radiation wavelength dependence with that of a black-body source, as represented by 15P1(A, T), implies that materials shock compressed to ~70 GPa are dominantly thermal radiators (Lyzenga et al. [11], Boslough [10]). At lower pressures, however, most materials apparently radiate both thermally and nonthermally (Si02: Kondo and Ahrens [12], Brannon et al. [13], Schmitt et al. [14]). Note that several of these materials are initially dielectric (e.g. Si02). The microscopic processes responsible for the radiation from these materials (defect electronic transitions?) are presently unidentified, but may be implied by spectrometric observations (Kondo and Ahrens [12]). In principle, energy transport in the target occurs by both radiation and conduction; our task is much more difficult if both radiation and conduction contribute equally to this process. In simple terms, we can estimate the likely relative contribution of radiation and conduction to energy transport within layers and across interfaces via dimensional analysis. The relevant nondimensional number is known as the Stark number, Skr (Siegel and Howell [15]), referenced to some state r, and given by (eq. D.15)
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Sk
=
r
339
krlXTrIXr arxrnrusaTr
This number represents the ratio of conductive to radiative flux, whether across a layer or within an “infinite” medium. It is composed of the materal properties kr~a~and n~,i.e. the thermal conductivity, radiation absorption coefficient, and refractive index, respectively. The remaining parameters, which may be material, include x~,Tr and z~ T1, the governing length scale, a reference temperature and 2 K4 is the Stefan—Boltzmann constant. temperature range, respectively. ~ we5.6696 10~W/m For the layered geometry of theAlso, target, may xassociate Xr with the layer thickness, Tr with the compressed or released temperature of the layer, and ~ Tr with the change in temperature across a given layer in the target such that L~Tr/Xrreflects the magnitude of the average temperature gradient across the layer. Via Skr~ we see that radiative transport dominates conductive transport in (1) an optically thick (a~—*oo), (2) a poorly conducting (k~—*0)and/or (3) a high-temperature medium, all other parameters being finite. Applying this parameter to the balance of energy in a target consisting initially of a metallic DP and FL (e.g. Fe), and a dielectric TW (e.g. A1 3 for the SW and the we find (appendix that 5k,. D). 10 and —i0 (compressed/released/reshocked) DP or FL,203), respectively In addition, viewing each layer as an infinite medium implies that conduction affects the balance of energy in both the FL and SW over a length scale of =
—
4. Initial conditions for conduction and radiation Within the assumption that (shock) compression, release, and reshock affect the temperature field on a much shorter time scale than conduction, these former processes establish the initial temperature
340
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
distribution in the target. Consequently, we begin constructing the model by estimating the temperature changes in each target component due to compression, isentropic release, phase transitions and/or reshock.
4.1. Shock-compressed state
For a given target material, let
{T1, P1, v1, p~,s1, e~} represent its uncompressed, or initial thermodynamic state, as characterized respectively by the temperature, pressure, material velocity, mass density, specific entropy, and specific internal energy of the material in that state. Likewise, let 5H’ eH} represent its compressed, or Hugoniot, state. For most (if not all) shock compression {TH, P11,experiments, VH, PH’ we have v~= 0 m/s, T 1 = 298 K and P1 = 0.1 MPa. With the constitutive assumptions that each target material (1) compresses adiabatically and (2) compresses as an ideal fluid, we obtain, in part A.1 of appendix A, the balances of mass (eq. A.2), linear momentum (eq. A.22), and energy (eq. A.30) across the shock front, u
=
—
~[‘(v11
—
v1)
.~
(4.1)
,
(4.2) —
e1
=
(2pI)~(PH+ P1),
(4.3)
respectively, where ij 1 PI’PH is the relative compression, (u v~)is the shock front propagation velocity (with respect to the uncompressed state), and u is the shock front displacement (i.e. “absolute”) velocity. With the further constitutive assumption that (u v1) is a linear function of —
—
—
(VH
—
(4.4)
u—v—a1+b1(v11—v1)
(e.g. Rice et al. [20]), and the initial state known a priori, (4.1)—(4.4) represent 4 equations in 5 unknowns: PH’ u, VH, P~’and ef{. In (4.4), a~and b1 are, respectively, the intercept and slope of the U~VH relation referenced to p1 and v~.Substituting (4.4) into (4.1), first for (u v~)and then for (vH v1), and then substituting the resulting two expressions into (4.2), the balance of linear momentum across the shock front takes the form (McQueen et al. [21]) —
—
2. (4.5) P1 = p1a~i~/(1b1ij) Relation (4.5) is known as the shock wave equation of state. The validity of (4.5) is clearly limited by the validity of the constitutive assumption (4.4), and the fact that (~H F 1) as b1 ~ 1. U—Va data for Fe (Brown and McQueen [22]) and many other materials (see Marsh [23]) appear to be well fit by a linear relation between u and VH. Relations (4.1)—(4.4), and so any relation directly derived from them, such as (4.5), are independent of any thermodynamic considerations, such as thermodynamic equilibrium and phase transitions. In particular, if compression induces a phase transition of some phase a, stable at T~and P°,to some other phase 1~,stable at T~and P~,we may write (4.1)—(4.5) in the forms —
—
—
—~
—+
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
341
~
(4.6)
P~ P~° = p~°(u v~)(v~ ~:°), —
—
—
(4.7)
—
t°(s~’, p~’)=(2p~)1’q~(P~+ Pr),
(4.8)
e~ase~(s~, p~)—e
~
(4.9) a
a
a
a2
2
‘~H’~1=pj(aj)~~/(1—bji~~) , (4.10) respectively, where ~ as 1 p°/p Although shock compression itself is a non-equilibrium, irrevers—
~.
ible process, it is still possible that (4.6)—(4.10) connect states of thermomechanical equilibrium, i.e., that the initial and Hugoniot states represent states of uniform, constant temperature, pressure, chemical potential and material velocity. In this context, we may connect the initial and Hugoniot states as distinct equilibrium states via a classical thermodynamic path (i.e. a path connecting a series of equilibrium states) representing a change in specific internal energy equal to that determined by (4.8), the balance of energy across the shock front, as follows (McQueen et al. [21], Ahrens et a!. [24]). Define z~eP0aseP(sp,p~) e’°(s~°, p~0)aseP e° —
(4.11)
—
as the difference in specific internal energy between a at T~and P~,and 13 at T~and
~p.
Since all of the high-pressure properties and fields appearing in the relations to follow in this section belong to 13, we drop the 13 superscript on these for simplicity in what follows. If we compress 13 isentropically from its initial density pf to its Hugoniot density PH’ the change in its specific internal energy is
~eSHase(sP,pH)_e(sP,pP)__f~
P 5(p)dlnp,
(4.12)
where P5( p) as P(s~,p) is the pressure as a function of density along the 13 isentrope referenced to s~. Since this last state and the Hugoniot state are at the same density PH’ we may connect them by an equilibrium thermodynamic path at constant density or volume. With de = T ds = c~d T at constant density or volume, where c~,is the specific heat at constant volume of 13, define
~eVH e(sH,
PH)
—
e(s~,PH)
=
J
cvH(T) dT
(4.13)
T011
as the difference in specific internal energy between 13 states at the same density PH~In (4.13), c~/54is the specific heat of 1~at a density PH’ TSH is the temperature of the 13 state at a density PH along the 13 isentrope referenced to s~,and TH is the temperature of the 13 state with specific internal energy equal to (4.8). TSH may be obtained from the relation (c9lnTIdlnp)5=y,
(4.14)
342
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
when given a knowledge of y, the equilibrium thermodynamic Grüneisen parameter of (4.8) and (4.11)—(4.13), we obtain
TOH
cvH(T) dT = —(~e~~ +~
+
fla(~H
Combining
(4.15)
Pr),
+
13.
a relation for the equilibrium thermodynamic temperature of a state of 13 with specific internal energy equal to that of its Hugoniot state. If we assume that y of 13 is a function of density alone (e.g. Boehler and Ramakrishnan [25], Anderson [26]), then the relation py(ôe/ôP)~= 1 for 13 at PH yields =
PHYH with
~
as
TSH=
(4.16)
(~H — ‘~sH)’
PS(PH) and YH
as Y(PH).
In this case, we also have from (4.14)
TPexp(fy(P)dlnp).
(4.17)
Combining (4.8), (4.11), (4.12) and (4.16), we obtain [1 (1 + 2YH)fla](PH —
—
‘)~)= (1
—
71co)(1~sH—
‘)~) p~yH(~eI+ ~e5~), —
(4.18)
which is similar to the form used by Jeanloz and Ahrens [27]. Both (4.15) and (4.18) represent the balance of energy across the shock front, assuming that the compression process connects states of thermodynamic equilibrium. However, since (4.18) depends upon the assumption that y is a function only of density, while (4.15) does not, these two relations are not completely equivalent. We use (4.15) to calculate T11 as a function of or PH, once we have relations for ~ zXe511, cr11 and TSH. Recall that, to obtain (4.18), we compressed the material from a state to ~H’ incurring a phase transition in the process. Imagine now that we can compress 1~from a state ~ to ~ For this case, (4.1)—(4.5) become 1(vH—vP), (4.19) ~H
.~
u—vP=n~ —
P~= p~(u v~)(v —
11 v~), —
—
e~= (2pP)~~(PH + Pr),
(4.20) (4.21)
(4.22) 2m~/(1 br~)2, (4.23) pr(ar) respectively, with as 1 PP/PH. Relations (4.19)—(4.23) represent a generalization of the “meta—
—
=
—
B.
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
343
stable” Hugoniot concept of McQueen et a!. [21] to an arbitrary initial state Z~.We emphasize that = Z~is by definition the same state in both (4.6)—(4.10) and (4.19)—(4.23). Combining (4.8), (4.16), and (4.21), we obtain [1—(1+ 2yH)fl13](PHPI)—(lTh3)(PSHPI)pIyH~eSH,
(4.24)
which is analogous to (4.18), and may be obtained from (4.18) when we replace p°with p~,P~with P~, and set ~ = 0. Noting that
2(ôe/e~p)f= p2(d z~e~/dp),
(4.25)
P(sf, p) p (4.24) may be rearranged to give as
PH(~)
—
YH AeSH = ~j
[1 (1+ 2YH)~l13](PH —
P~)+
—~--
(4.26)
P~,
i.e. an ordinary differential equation for a a a l3 13 z~esH—1Xesu(pH;pi,a1 ,b1,p~,‘y1),
when P~is given by (4.10). We solve (4.26) numerically, subject to the initial (1) ~eSH(pP)= 0 [by definition, i.e. (4.16)], and (2)
(PH
=
p~)conditions that
p~(dL~eSH/dPH)~ ~ = from (4.25). Note also that consistency between (4.18), as based on (4.6)—(4.10), and (4.24), as based on (4.19)—(4.23), requires that as
P5H(PP)
2= P =
PH(pP)
=
P~+ (ib~)
1
+
2p~~
(4.27)
with ~ as 1 p~0/p~~ Clearly, assuming p~ p~° and ~ 0, we have P~ P~°. The change in specific internal energy along the 13 isentrope referenced to s~,AeSH, is usually calculated via some “equation of state” P(s~,p) for 13. In most cases (e.g. finite strain: Stacey et al. [28]), the resulting relation for ~eSHtakes the form —
=
~eSH(pH,
p~,K~,K~’),
where K~and K~’are the isentropic bulk modulus of 13 and its first pressure derivative, respectively, referenced to Z~.In fig. 4.la we compare values of i~e0~ for r-Fe calculated via (1) eq. (4.26), (2) third-order spatial finite strain (e.g. Stacey Ct a!. [28]), (3) third-order Ullman—Pan’kov finite strain (Uliman and Pan’kov [29]), and (4) Murnaghan’s equation of state (e.g. Stacey et al. [28]). The parameters used for this calculation are listed in table 4.1. As noted by Somerville and Ahrens [30], the third-order Ullman—Pan’kov relation is compatible with the linear U~VHrelation (4.4), since both make similar predictions for higher-order derivatives of K~.Again, note that (4.26) and methods (2)—(4) give
344
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
0
90
a
180 270 Pressure (GPa)
360
0 0~
(S —
—
—
Third-order-FS UlImoo-Poo’kov
‘ ‘.‘ .
Energy-balance
‘ ,‘ —
,,,
~rnoghon1~
0
90
180 Pressure (GPa)
270
360
Fig. 4.1. (a) Comparison of different estimates of the change in specific internal energy (SIE) of c-Fe as a function of pressure along an isentrope referenced to ~ ~. These calculations are based on the parameter set for c-Fe given in table 4.1. “Energy balance” refers to eq. (4.26) in the text. (b) Comparison of different estimates for pressure along the isentrope, ~aH’ of c-Fe referenced to Z ~, based on the parameters for c-Fe given in table 4.1.
us
z~eSH= ~eSH(pH;
~
a~°, b~,p~,‘yr) and
~e
5~
=
~eSH(PH; p~,K~,K~), respectively. For con-
sistency and a true comparison, we constrain the values of K~and K~for e-Fe via the relations 2, (4.28) K~= p~(a~) when we assume that a~is equal to the bulk elastic wave velocity, and (Ruoff [40],part A.4 of appendix A) K~’=4b~—1,
(4.29)
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
345
Table 4.1 Standard temperature and pressure~~ (5TP) parameters Quantity 3) p (kg/rn a (m/s)
c-Fe 8352’~
liquid Fe 7952’°
Al
4487’~ 1.57’~ 168c)
4038’~ 1.58’~
8908° 0.91° 254~~
203
LiF
39~C)
2650d)
K (GPa)
5.28) cj,, (J/kgK) 1) ya (X10’ K q t,~(K)
130g)
5.31’~
432°~
~5i) 43k)
7750) 1.6~
1.95m)
L32m)
i.o’~
i.ø°~
1250’° 2345~
385D
TM (K)
18090)
k (WImK) 80~ i~(nfl m) 50°~ *) The reference pressure for a~and b~, and so for
46p)
0
K,
and
5050d)
~g)
4.28~ 10.3~
1615d)
1.78w)
580d)
845°~ 3p)
0
K~,of c-Fe and liquid Fe is actually P~,as
given by (4.27). 5T)(Drotning [32]). a) Jephcoat et al. 1311. b) Robie o) Calculated et al.from [33]. p(T) = 8136(1 — 7.608 x 10~ d) Van Thiel [34]. Estimated from u 3955 + l.S8vH (Brown and McQueen [22]).
=
Fit to Al
°
203 data in ref. [23]. ~ Calculated assuming K, = ~ 5) Anderson et al. [35]. ‘~Calculatedwith K~ = 4b, — 1 (Ruoff [40]). ~ Andrews [42]. ~Assumed the same as a-Fe (Touloukian et al. [36]). °
Touloukian et al. [36].
o)
Weast [37], p. D-187.
w) Calculated from y = aK,/pc,.. n) yp = const. assumed in all calculations. ‘~ Assumed
the same as a-Fe in Touloukian et al. [38].
q) Inferred from Keeler [39]. assuming only that 13 compresses as an ideal fluid. To use (4.28)—(4.29) for this purpose, we require values for a~ and b~.Again, since both (4.10) and (4.23) give us ~H = P~,via the definitions of a~and b~as coefficients in the metastable U~~VHrelation (4.22), we may use (4.10) and its density derivatives to estimate a~and b~,as shown in appendix B. From this procedure we obtain a~and b~ as functions of p~°, a~,b~and p~in the limit PH~~~PP. Leaving the details to appendix B, we have a~ (1 1 —
4
—
3]u2a~ , ~)[(1 + b~’~)/(1 b~~) 4(1— ~ (1- b~)(1+ b~)
(4 30)
—
which are (A.64) and (A.65), respectively. Note that, from these relations, af—* a°and b~—~ b~as p~—~p’, i.e., as t~—~0. With p~known a priori, and a~° and b~constrained from experimental U~~~VH data, then, we need only p~to obtain a~and b~from this method. The calculations presented in fig. 4. la demonstrate that ~e 5~predicted by each of these relations
B. Suendsen et al., Optical radiation from shock-compressed materials and interfaces
346
(except that of Murnaghan) is essentially the same. Note that the Murnaghan isentrope is off scale in fig. 4.la. Once we have ~ via solution of (4.26), (4.24) provides the “equation of state”, i.e., ‘~sH =
P~+ PHYH z~eSH+
[1—(1+ 2YH)’7’31(~H Pr). —
(4.32)
For the derivatives of ~eSH, of course, the minor differences between the different expressions for z~eSH are magnified, as shown in fig. 4.lb, where we plot ~sH as given by (4.32), and compare it with the values for ~sH given by the equivalent finite-strain and Murnaghan relations. In particular, note that the energy balance equation of state (4.32) is a bit “stiffer” than either of the finite-strain equations of state.
Since both dielectric and metallic solids initially compose our target, we must consider thermodynamic properties that reflect the influence of both ionic and electronic processes. To estimate the harmonic lattice contribution to these properties, we use the Debye model (e.g. Alt’shuler et a!. [41], Andrews [42]). Andrews [42] used this model as part of a parameterization of the equilibrium thermodynamic properties of a- and E-Fe. Jamieson et al. [43], Brown and McQueen [44, 22], and Boness et a!. [45]have all assessed the influence of electronic processes on the material properties of metals at high pressure and temperature. They assume the conduction electrons contribute to the equilibrium thermodynamic properties of a metal as a Sommerfeld free-electron gas (e.g. Wallace [46], section 24). This is reasonable for for T much than thethan Fermi Since this for 5 K, assuming that T~ Fe is less much less thetemperature. Fermi temperature of temperature Fe seems quite Fe is =i0 reasonable for the calculations presented below. Assuming that lattice-electron and band-structure contributions are negligible, the molar Helmholtz free energy (HFE), F(T, p), of a cubic or isotropic solid material subject to an isotropic state of stress is then given by (Wallace [46], section 5) F(T,p)=cP(p)+3PR[~D+ln(1—e~°)—~ED(~D)]T— ~1T2(p)T2.
(4.33)
Here 1(p) represents the zero-temperature lattice contribution to the molar HFE, =
~D(T, p)~eD(P)/T
(4.34)
is the Debye parameter, (l(p) as T(p)
—
2A 2(p),
(4.35)
2is the low-temperature (Tmuch less than the Fermi temperature) electronic contribution to F, ~F(p)T A 2 is the high-temperature anharmonic contribution to F, v is the number of atoms in the 2C°)formula, T chemical and ED(~)is the Debye internal-energy function (e.g. Gopal [47]), —
ED(~)~4JxXl
dx.
(4.36)
In the Debye approximation, the Debye temperature °D is related to a lattice Grüneisen parameter, YD’ via (Wallace [46])
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
347
(4.37)
YD =(dln OD/dln p).
If we assume
(4.38)
, YD(P) = YD(P~)(PIIPYt°
then OD is given by @D(Pi)exP(q
~D(P)
[1_~/D]).
(4.39)
For simplicity, we do not separate YD into longitudinal and transverse components; in this case, YD represents a weighted average of these components. The quantity E(p) is related to the electronic Grüneisen parameter, Ye’ via (Wallace [46]) Yeas —(din F/dln p).
(4.40)
Assuming Ye is constant, we have F(p)asF(p~)(p (4.41)
1/p)~. By analogy with (4.41), we assume for simplicity that
(4.42)
M. (1(p) as
(l(p1)(p1/p)
For the example calculations involving Fe targets presented below, we constrain the values of (1( p~)and w empirically. Boness et al. [45] calculated F( p) for the 6- and -y-phases of iron using the Sommerfeld free-electron gas theory. In addition, these authors suggest that the electronic density of states in liquid iron at high pressure may be approximated by that of the close packed 6- and y-phases at high pressure, where the liquid should be “close-packed”. We make the same assumption. Relation (4.33) allows us to write expressions for the approximate density and temperature dependence of a number of solid-state properties (appendix B). In particular, the equilibrium thermodynamic Grüneisen parameter, y, based on (4.33), is given by (eq. B.10) (4.43)
c~
where M is the molecular weight and cv the specific heat at constant volume. From this, we note that y is only weakly temperature dependent, since w YD in the pressure range of interest. Hence, the thermodynamic model based on (4.33) is approximately consistent with the assumption above that ‘ is a function of density alone, upon which (4.18) above is based. Consequently, we assume ~v YD(P)’ where YD(P) is given by (4.38). In this case, from (4.14) and (4.38), we also have T(s 1, p2)
=
T(s1, p1) exp( YD(Pi)
[(~/~)~D
—
(p./p2)~D])
(4.44)
348
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
for the change in temperature in the material due to isentropic compression or expansion from a starting density p1 to a second density p2 along the isentrope of the material referenced to s1. With respect to T~,we are particularly interested in the specific heat at constant volume, c~(T,p), given by c~(T,p)~—~(-~~)=~~ [4E1~(4~)—
~~]+
~
T.
(4.45)
Substituting this into (4.15), we obtain [E~(~1.~)T~ED(4D3)T$H] —
+
~-~,j(2H(T~
—
(4.46)
T~H)= ~eVFI,
12(PH). Note that i~eVH= 0 when TH = T with (l~ 311 and that TH> T311 when ~ >0. In (4.46), we use ~Ds ~DH’ TSH, ~DH ODH/TH and °DHas OD(PH). Equation (4.46) is an implicit relation for TH, which we evaluate numerically. Since the majority of our calculations are at high temperature (T> ~D), and the Hugoniot temperature changes much more drastically with pressure than ~ we 4DH—~O)form, may approximate ED(~DH) by its high-temperature ( fDH
ED(~DH~0)=(~3) 3 ~ ~
(4.47)
Substituting this into (4.46) and rearranging, we obtain
(4.48)
~
0DH + ~
~VH
+ ~
(4.49)
u2HT~H.
ED(~DS)TSH+ 9vR
Relation (4.48) has the solution TH=2v’~cos(~ cos1(q/pV~))—2vR/(2~,
2vR2 2M p=—(-~-—)+~-n—nH’ H
H
~
yR
2M
H
H
(4.50) 2
2vR2 H
If we set (l~equal to zero in (4.46), we have ED(~DH)TH((2H= 0) = ED(~DS)TSH+ ~
~
(4.51)
which is appropriate for a dielectric material with negligible anharmonic contributions to F. Doing this in (4.48), we obtain
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
M 1 3~R~9 21/2 TH(flH=0)=~—~A41+[1-~ MA~)]
(
}~
349
(4.52)
fll~as~evH + 3yR ~j_ ED(~D)TSH + 9~R ~ ~DH~ Relation (4.52) is appropriate for a dielectric material at high (T>
9D)
t
temperature with A
2T~H< A2T~‘~ 1. Alternatively, if we assume the classic limit for the harmonic contribution to cv at high temperature, i.e. 3vR/M, eq. (4.46) reduces to 3vR TH(~DH=0)=—fl --—
M12 2A~) LY~2~ (~R) [1’
\h/2
A~asi~ev~+ ED(~DS)TSH+ ~
—1], 1
(4.53)
QHT~H.
_~_
Finally, if we set both f2~and ~DH equal to zero in (4.46), we have TH(QH
=
0,
4DH =
0)
=
ED( ~Ds) TSH
+
3R z~e
(4.54)
5,54.
If we further assume ED(~D$) 1 in (4.53), we obtain the relation most commonly used (e.g. Jeanloz and Ahrens [27]) to calculate TH. To demonstrate the effect of these different approximations to Cv on TH, we plot TH as a function of pressure for Fe compressed from a-Fe to s-Fe in fig. 4.2, using the parameter set for Fe given in table 4.1. Below, we constrain fl~from the intersection of the Fe Hugoniot and melting curves, but for the purposes of this calculation we assume that A2 = 0 and use the results of Boness et al. [45] for F(p) (table 4.1). From fig. 4.2, we see that (l~is the dominant influence on the Hugoniot temperature of 2-Fe at high pressure, and that the difference between TH calculated with the full Debye relation for cv, 0 C,
0 C
Do
Debye-plus-electrons Debye-minas-electrons
0
a
DnIong-Pet~t-p1us-eIectrnns
-
co Du1ong-Petnt-~nus-elec1rnns
-~
~
0 0~ 0
Q)~t
—
___
a—
-
0
1
Melt,ng —
—
0
in
.—
1o
-
~1l~i
E~
0
60
120
180
240
Pressure (GPa) Fig. 4.2. Comparison of the effect of different models for the specific heat at constant volume, c~(T,p) on the temperature of the c-Fe Hugoniot states.
350
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
eq. (4.46), and TH calculated by assuming the harmonic part of Cv = 3vR/M, eq. (4.53), is very small (about 200 K at 240 GPa). In fact, even for Al2 03, which possesses a much higher Debye temperature than 6-Fe (table 4.1), Hugoniot temperatures calculated with eq. (4.51) are only about 300 K above those calculated with eq. (4.54) at 200 GPa. Note that the curves in fig. 4.2 converge at low pressure because T~approaches TSH “faster” than the various approximations to c~can affect TH as P~—~ P~. To summarize, from the impedance match and u—vu relations of each material, we may obtain the pressure and density of the first shocked state of each target component. Using this along with an estimate of ~ePh1 and the assumed form for ‘ above allows us to estimate, using eqs. (4.26) and (4.44), the changes in specific internal energy and temperature along the appropriate isentrope of the high-pressure phase of each target component. These estimates, along with the model (4.45) for c~(T,p), allow us to calculate TH for any phase as a function of P~or p11. The next step is to estimate the effect of release on the compressed state (TH, P~and PH). 4.2. Release and reshock states As discussed above, the targets are constructed so that the DP has a higher shock impedance than the FL, which in turn has a higher impedance than the TW. In this case, both the DP and FL are compressed, released and possibly reshocked. Assuming that a given release state of the DP or FL is in thermodynamic equilibrium, we may again employ the concept of an equivalent equilibrium thermodynamic path to connect respective compressed and released states of each target component. However, since we have no expression for the change in specific internal energy of the material during release that is independent of the details of phase transition [recall that eq. (4.3) is such a relation for shock compression], we cannot utilize the same kind of equilibrium thermodynamic path as that constructed above for shock compression. Instead, we must assume something about the release process, and any potential phase change during release, to construct an equilibrium thermodynamic path between the compressed and released states. The only constraint we have a priori is that the release process takes the compressed material at an interface, via the release path, to a state with approximately the same normal components of material velocity and stress as the compressed state of the lower impedance material on the other side of the interface. Subsequent wave reverberations establish the continuity of normal stress and material velocity, as required for the existence of a material interface. To proceed further, we assume that heat transport in or out of the target is insignificant on the time scale of the release process, i.e., this process is adiabatic. Considering each compressed target component as an equilibrium thermodynamic system, we further assume that any mechanical work by the system during release is entirely reversible. In the case of a single-phase system, the release path is then both isentropic and adiabatic. The change in temperature with density along this path is related to y, as given by (4.14) above. Since the impedance match provides us with the pressure of the release state, ~R, we may calculate the temperature, TR, and density, PR’ along an isentropic path that has not encountered a phase boundary, through simultaneous solution of (4.14) and
p(TR,
~R)
=
p(THR,
~R)
exp(_
f
a(T, p(T, ER)) dT),
(4.55)
THR
where TR
T(sH,
PR)
is the release state temperature,
PR as p(TR, PR)
the release state density, and
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
351
p( THR, PR) is the density along the (possibly metastable) Hugoniot of the same phase at a temperature THR and the pressure of the release state, ~ The coefficient of thermal expansion a( T, p) in (4.55) is derived from an equilibrium thermodynamic model for F( T, p) of the appropriate phase (appendix B for solid state, Svendsen et al. [48] for liquid state), since by definition (Wallace [46]) 2F/ôT9p)PT
2
aas 2p(9Fh9p)~+p p(ô (92F/ôp )T
(4.56)
as,
KT
where KT = KT(T, p) is the isothermal bulk modulus; for the solid-state model, it is referenced to the isentrope or Hugoniot, as discussed in appendix B. To bound the nature of the release process as initiated at an interface, we focus on the extremes: (1) complete contact (roughness ~ shock front thickness) at the interface, or (2) no contact, in which case each material has a free surface at the interface. We refer to the former interface as the “smooth” interface, and to the latter as the “rough” interface. To illustrate the different paths these “endmember” interfaces may take, consider the two examples discussed below and depicted in fig. 4.3. If we compress the DP (respectively, FL) to some point A along its Hugoniot below the Hugoniot—melting curve intersection, it will release to a state having, after one or two wave reverberations, the normal stress and material velocity of the compressed FL (respectively, TW). If these reverberations are isentropic, the resulting temperature will equal that calculated by direct release to the pressure of the compressed FL (TW). The compressed DP (FL) material at the smooth interface then releases directly to this state, represented by point B in fig. 4.3. However, the surface of the DP (FL) at the rough interface is partially free; hence, the compressed DP (FL) material at this interface releases to near-zero pressure. If we assume the release path is isentropic, its slope will be less steep than that of the melting curve. In this case, the melting curve and release path will intersect (point C, fig. 4.3). If the phase transition is slow relative to the rate of decompression, the releasing DP (FL) material will follow the (metastable) path ABCG to low pressure (even though this path is not necessarily isentropic, as discussed below). However, if the transition is uninhibited, the release path will turn along the phase
LIQUID
SOLID
Pressure Fig. 4.3. Possible range of temperature—pressure paths taken by DP and FL materials near the DP—FL and FL—SW interfaces, respectively, during an experiment.
352
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
boundary at the intersection point (point C, fig. 4.3), and the mixed-phase material will decompress along the boundary until the transition is complete, or the mixture reaches the release state. Assuming the transition completes above zero pressure at joint D in fig. 4.3, the now liquid DP (FL) leaves the phase boundary and continues to decompress along DE to P1. As the released DP (FL) material closes the interface, it impacts the FL (TW) material and is reshocked and reverberated along a series of paths, collectively symbolized in fig. 4.3 as the paths lying between EF and GH, up to the smooth-interface, release-state pressure, which is that of the shocked FL (TW). Note that the temperature achieved by this set of shock paths is bounded above by the temperature estimated from a single compression back up to the Hugoniot pressure of the FL (SW); we use this bound, along with isentropic release, to estimate reshock temperatures in what follows. Since the temperature of the released DP (FL) is much higher than the initial temperature of the unshocked DP (FL), the temperature of the reshocked DP (FL) material is higher than either the first shock state or the release state of the DP (FL) material at a smooth interface (point F versus point B, z~TRs,fig. 4.3). If the DP (FL) material is shocked to a higher pressure state than A but still below the melting curve, a smooth interface may release to a state pinned to the melting curve, or be above the melting curve, as for release from A’ in fig. 4.3. The rough interface released from A’ would follow A’B’C’ and be reshocked along C’D’ to D’. Note that the effect of reshock is much more pronounced as the pressure of the first shocked state increases, regardless of the phase transition. When the release path encounters a phase boundary, such as the melting boundary shown in fig. 4.3, eq. (4.14) is no longer valid. If we believe the release path remains isentropic through this region, then we must require that, in addition to releasing adiabatically and doing or experiencing only reversible work, the material also change phase in thermodynamic equilibrium (see appendix C). Under these conditions, the isentropic two-phase path for a congruent phase transition from phase 13 to phase ‘~ is described by the relation [(c~ + x ~c~) ~v
13v’° + x ~av)T~
—
8 ~s] dP + iXs TPB ~ d,~’= 0,
(a
(4.57)
where P is the pressure along the phase boundary TPB(P), x is the mass fraction of ii, c,,, is the specific heat at constant pressure, a isanythequantity coefficient of thermal expansion, v is Since the specific and is the jump of ~(‘ across the phase transition. (1) all volume, end-member quantities in (4.57) may be viewed as functions of pressure and temperature, and (2) temperature and pressure are not independent along the equilibrium phase boundary, these quantities are actually a function only of pressure or temperature along the boundary. In this case, choosing P as independent, we may solve (4.57) for x = ~(P) to obtain (eq. C.29) 13
—
~i
~(P)= ~(P)X(P)(~(PH)x(PH)x(PH)+
I
~(P*)(T
av ~s
—
c’3p ~v) dP*),
(4.58)
PB
P
liii
p~(P)~exptJ
—
Lz~av
—
~v/ —
TPB
(~1+
~c\1 ) —~--~)j dP*
1,
(4.59)
and X(P) ~s TPB(P) ~s. In (4.58), x(PH) is the mass fraction of rr (e.g. liquid) present in the Hugoniot state. We evaluate (4.58) numerically in the mixed-phase region along the phase boundary TPB(P) until (1) ~(P) + x(P11) = 1 (complete transformation) or (2) P = ~R (partial transformation). In
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
353
the former case, the new phase then releases to ~R along a path beginning at the pressure and temperature on the phase boundary where ~(P) + X(PH) = 1. Relation (4.58) is valid along any isentropic path through a first-order mixed-phase region of a single-component system (i.e. solid—solid, liquid—solid), but now we focus on the solid—liquid phase boundary as discussed above. To utilize (4.58), we need relations for the solid and liquid state properties along the solid—liquid boundary TM(P). We do this by way of semi-empirical models for the solid state (e.g. Andrews [42], and appendix B), and the liquid state (e.g. Stevenson [49], Svendsen et al. [48]), respectively. For the solid state, we use a parameterization of the solidus, TM(p~),based on Lindemann’s law, -
(dTM/dP~)Lindemann =
(4.60)
T~A~/p~,
where A~as2(ysM_~)
(4.61)
for the solid—mixed phase boundary (solidus). The quantity ~ is the solid phonon Grüneisen parameter at the melting point, equal to YD(P~)in the Debye approximation we use here. Using (4.61) in (4.60), we may calculate TM(p~)once we know the density of the solid along the phase boundary, and the dependence of A~on the solidus density. For the solid phase, we have already assumed as YD(P)’ with YD given by (4.38) above. Putting (4.38) into (4.61), and the resulting combination into (4.60), we obtain an expression for the solidus, i.e., TM(p~)as TM(p~M)(~~)exp{~~2~r [(-~)h1r
(4.62)
—
where p~11is the density of the solid at the melting temperature at standard pressure. To use (4.62) to find TM(p~),we need to calculate the change in density along the solidus with pressure. Noting that, since in general the equilibrium thermodynamic properties as developed from (4.33) are functions of temperature and density, we may calculate the variation of any of these properties i~r(T, p) with temperature at constant pressure from the relation ~(T, P~)= ~(Tr, Pr) +J(~)dT,
(4.63)
(d~/iIaT)~ = (9~~IaT)~ap(ôçfJIdp)~,
(4.64)
—
and ‘~rand Tr are some pressure and temperature at which we know cu. In particular, putting = p into (4.62), as we did to obtain (4.55) above, we may anchor the solidus TM(p~)to the solid Hugoniot of the relevant solid phase by solving (4.63) (numerically) simultaneously with ~i
p~(P)=
P(TH,
P) exp(_f a(T, p(T, P)) dT),
(4.65)
B. Svendsen
354
et al., Optical radiation from shock-compressed materials and interfaces
C
C
_____
C
C C ~
C
C -
CC -
=
C C
- - -
C. C
“
.
-
-
- -
•1-
—
--
H -=
C C C —
C ___________
0
90
—
oolidus-compression Hugoniol-compression
_________________
180 Pressure (GP)
270
360
Fig. 4.4. Comparison of c-Fe Lindemann solidi calculated from the compression along the Hugoniot (dotted curve) with that estimated from the compression along the Hugoniot adjusted to the solidus temperature (dashed curve), as discussed in the text.
where p(T~,P) is the Hugoniot density of the solid phase at the same pressure. In fig. 4.4, we compare this calculation with one in which we assume P(TM, P) = p(TH, P). The greatest effect is at low pressure; this is also where the correction is most uncertain. For the rough DP—FL and FL—TW interfaces, the DP and FL release completely, and consequently we cannot use the Hugoniot as a reference state. So instead of (4.55), we solve (4.14) simultaneously with P(TR,
0)
=
P(Tr,
0)[1
—
a(Tr, 0)(TR
—
Tr)],
(4.66)
where Tr is some reference temperature (e.g. 298 K or TM), depending on the relevant phase. With the density of the release state, PR’ we may estimate the free surface velocity of the DP and FL surfaces at the DP—FL and FL—TW interfaces, respectively, due to isentropic release, via the Riemann integral method (Rice et al. [20]). We assume, as required by the constraint of isentropic release, that the material velocity is continuous across the phase boundary (i.e. the same for both phases) when calculating the free surface velocity. We then take this free surface velocity as the “projectile” velocity of the DP (FL) surface impacting the FL (TW) surface, and use an impedance match to calculate the pressure and density of the reshocked state. To calculate the temperature of the reshocked state, we use the appropriate form of (4.46), but referenced to the temperature and density of the complete release state rather than to T,~and p~. 4.3. Application to Fe targets To exemplify these considerations, we calculate release and reshock states for Fe—Fe—A1203 and Fe—Fe—LiF targets, as shown in figs. 4.5a,b. The solid and liquid T~,P11 states result from (4.46) and (4.53) with A2 = 0 and ~Ds = 0, respectively, as based on the parameter set given in table 4.1. Solid-state properties along the release path and melting curve are referenced to the r-Hugoniot via
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces C C 0 I
C .x,
~
—
Lindemana-melting
aa
Hugarnot-states Release-states
00
o
::
::~::i:::eton
~
Foil-date
0
-
000
~
o
0
I
Relesse-eonduct,on
-
355
Liquid
00 000
80
160
240
320
Pressure (GPa) 0 0 CJ __
I —
— 11)
an
Hugoniot-atates
no
Reshoek-states
ax
Reshoek-conductian
~
Fail-data
0
ci
I
Lindemwnn-melting
Otis 00
Liquid
xi’,
0
0
x~
I~0t0tt1huit0annaaad
0
80
160
240
320
Pressure (GPa) Fig. 4.5. (a) Release/reshock calculations for Fe film/foil—Al 203 interfaces and initial greybody temperatures inferred from Fe—Fe film/foil—Al203 radiation data of Bass et at. [6]. “Release conduction” and “reshock conduction” symbols represent initial effect of thennal inertia mismatch across the Fe film/foil—Al203 interface on the indicated states. (b) Release/reshock calculations for Fe film/foil—LiF interfaces and initial greybody temperatures from Fe—Fe flim/foil—LiF radiation data. The larger shock impedance mismatch between Fe and LiF results in a lower release-state pressure at Fe—LiF interfaces than at Fe—Al203 interfaces, when both release from the same Hugoniot pressure.
(4.55), (4.62) and (4.65), while the analogous liquid-state properties are referenced to the highpressure Fe melting data of Williams and Jeanloz [50] (as reported in Williams et al. [51]) via a liquid-state model for Fe (Svendsen et al. [48]). In calculating release paths, we ignore all other solid phases of Fe, save r-Fe, which is the stable solid phase of Fe along its Hugoniot between 13 GPa (Barker and Hollenbach [52]) and ~—~200 GPa, where the sound speed measurements of Brown and McQueen [44,22] along the Fe Hugoniot suggest that 2-Fe transforms to y-Fe (?) or possibly a new solid phase 0 (Boehler [53]). Consequently, -y-Fe and / or another solid phase is in equilibrium with
356
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
liquid Fe above about 5 GPa to perhaps 280 GPa (e.g. Anderson [26]). In this case, we neglect any effects of an r -y or r 0 transition in referencing compression along the Fe melting curve to the 6-Fe Hugoniot. As stated above, in calculating the 6-Fe Hugoniot states shown in figs. 4.5a,b, we have constrained fl(p~), with w = 1.34, which is the value of ~ for 6-Fe given by Boness et al. [45], by requiring the parameterized Fe melting curve and 6 Hugoniot to intersect at 245 GPa. On the basis of the parameter set given in table 4.1, this fit constrains fl(pf) to be 0.046 i/kg K2. Boness et al. [45] calculated a value of 0.090 i/kg K2 for F(p~)(adjusted to STP density for 6-Fe given in table 4.1). If we set fl(p~) = 0.090, the 6 Hugoniot based on the parameter set in table 4i intersects the melting curve at =280 GPa. We note that Boness et al. [45] constrained F(p~)= 0.09 J/kg K2 and Ye = 1.27 for -y-Fe, while Bukowinski [54] constrained F(p~)=0.08J/kgK2 and Ye = 1.5 for this phase. With these values for F(p~),the value of fl(pf) constrained above for 6-Fe implies some competition between anharmonic and electronic contributions to the specific heat of 6-Fe at high pressure. Brown and McQueen [22] fit a linear u—vH relation to the available Fe Hugoniot data between 13 and 400 GPa. Since their sound speed measurements also suggest that Fe melts along the Hugoniot above about 245 GPa, their u—vH relation should describe the liquid—solid mixture and pure liquid phase, as well as the solid. On this basis, we use their u—v 11 relation to calculate both the 6-Fe Hugoniot and a metastable liquid-Fe Hugoniot referenced to the extrapolated density of liquid Fe at STP (table 4.1). With this u—v11 relation, eq. (4.53) above for TH (A2 = 0 and ~Ds = 0), and F as constrained by Boness et a!. [45], we calculate the metastable Hugoniot of liquid Fe. Using ~ = 0.14 Mi/kg for Fe (as compared to the enthalpy of melting at standard pressure, 0.25 MJ/kg, from Desai [55]), the metastable liquid-Fe Hugoniot intersects the melting curve at about 305 GPa. This agrees reasonably well with the results of Young and Grover [56], who also ignored all other phases of Fe save 2 and liquid in their parametrization of the Fe melting curve. We combine this metastable Hugoniot along with the 6-Fe Hugoniot in an ideal mix (e.g. Watt and Ahrens [57]) to construct the compressed, mixed-phase region shown in figs. 4.5a,b. For comparison with the calculations, we have plotted the initial (t = 0) interface temperatures inferred from the Fe film/foil radiation data of Bass et al. [6] in figs. 4.5a,b. Note that the Fe—A1203 interface data shown in fig. 4.5a run almost parallel to the reshock locus, thereby exemplifying the strong pressure dependence of the reshock process (Urtiew and Grover [7]). Comparing the data with the smooth-interface release states, shown as squares in fig. 4.5a, implies that Fe at both film—A1203 interfaces experiences up to —1800 K of reshock heating between 190 and 230 GPa. As stated above, we naively expected that the film—TW interface would experience consistently less reshocking than the foil—TW interface. The present results contradict this expectation. There appears to be no guarantee that film interfaces will consistently experience any less reshock than the foil interfaces, especially at high pressure. In this case, a well-polished foil surface may actually experience less reshock than a slightly porous film interface. Figure 4.5b displays the results of the calculation for Fe—LiF interfaces. Because of the larger impedance mismatch between Fe and LiF, the Fe—LiF interface reaches a lower release-state pressure than the Fe—Al203 interface, given that both release from the same Hugoniot pressure. The data and calculation imply that lower release-state pressure results in less extreme reshocking. Note that the Fe—LiF and low-pressure Fe—A12O3 data fall right on the corresponding smooth-release locus. The points labeled “release conduction” and “reshock conduction” refer to the effect of the contrast or mismatch in “thermal inertia” across the Fe—TW interface on the release and reshock temperatures, as discussed in the following section. —~
—~
B.
Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
357
5. Conductive transport in the target
We assume that the temperature profile created by shock compression, release and/or reshock is established on a time scale sufficiently short to represent the initial conditions for energy transport in the target. Urtiew and Grover [7] considered the problem of energy transfer at material interfaces and demonstrated that a rough (~1 lim) interface experiences a higher degree of shock heating than a smooth (~1 ~m) interface, much like a porous material experiences relative to its crystalline counterpart. Since the TW surface at the (FL—SW) interface is much less rough (~108m) than the FL surface at the interface, it should experience little, if any, direct reshock heating. However, the DP and FL surfaces at the DP—FL interface, as well as the FL surface at the FL—TW interface, may experience significant reshock heating, as discussed above. Following Grover and Urtiew [9], we assume that (1) energy transport is parallel to the direction of shock propagation (i.e. one-dimensional), (2) both temperature and heat flux are continuous across each interface in the target (appendix A.2), and (3) there are no sources or sinks of energy in any layer or at the interfaces between them. Under these conditions, we may solve the one-dimensional energy balance (e.g. Carslaw and Jaeger [58]) pc~dT/dt = k d2T/dx2
(5.1)
for the temperature profile T = T(x, t) in each target component as a function of position along the direction of shock propagation x and time t. The time t = 0 corresponds to coincidence of the shock front and the FL—TW interface. In eq. (5.1), p is the density, c 0,, is the specific heat at constant pressure, and k is the thermal conductivity. Since the temperature profile in the FL, and particularly the temporal variations of the temperature at the FL—SW interface, control the intensity of thermal radiation sources at the FL—SW interface, we emphasize these in what follows. We expect a layer of the DP material at the DP—FL interface and a layer of the FL material at the FL—SW interface to experience some degree of reshocking. Also, the rough FL surface at the DP—FL interface should compress into a thin layer with a much higher temperature than the compressed solid FL material. With this structure, the initial (t = 0) temperature profile in the shocked target is of the form (fig. 5.1) T(X,0)asTD, as TD+L~TDF,
—(d+ÔD)
asTF+E~TFD, —d
asTF+i~TFW, asT~,
—(d—ôFD)
(5.2)
ÔFW
0
Here, d is the FL thickness, while TD and TF are the temperatures achieved in the DP and FL, respectively, by direct release to the pressure of the SW, which temperature T~.Also, TD + ~while TDF 8D inhas thea DP at the DP—FL interface, isTFthe temperature of the reshocked layer with thickness + ~ TFW is the temperature of the reshocked layer with thickness 6FW in the FL at the FL—SW
358
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Driver
Film / Foil
I
I I 5) -d
I
I
~FW
~
I
I
-(d+8
Window
-(d-&~0)
x
Fig. 5.1. Initial conditions for thermal conduction in target. T0 and T,, represent temperatures achieved in the DP and FL, respectively, upon direct release to a state with the pressure of the SW, having a temperature T~.Variable degrees of reshocking are shown for DP (~T0~) and FL (~T~0) at the DP—FL interface, and for FL (~T~~) at the FL—SW interface; these involve some thickness (h0, b~0,and 6~~) of each target component adjacent to the interface.
interface. If the surface of the FL at the DP—FL interface is also rough on some scale, it will compress like a porous material. Consequently, we assume that a layer with a thickness ~FD and temperature TF + ~ TFD forms in the FL at the DP—FL interface. Since the surface of the TW is much smoother than the DP or FL surfaces, we assume that there is no reshock heating of the SW material at the FL—SW interface. Note that the DP and SW are idealized as thermal half-spaces, a consequence of our assumption about the rates of compression and release relative to conduction. Again, we emphasize that all material properties of each target component are assumed homogeneous and time independent, and are referenced to their respective states at the pressure of the FL—SW interface. The governing relation (5.1) for each layer, combined with the boundary conditions of continuity of heat flux and temperature, and the initial conditions (5.2), specify an initial-boundary value problem for T( ~, T) in the target, where ~ = x/d and T = t/texp~We are particularly interested in this profile for the FL layer (—1 ~ 0), and the temporal variation of T(0, T), which represents the FL—SW interface temperature. Solving this initial-boundary value problem in appendix E, we obtain [eqs. (E.22)—(E.25)] TF(~,T) = TF
+
A(~,T)I~TDF+ B(~,T)(TD
—
TF) + C(~,r)z~TFD
+D(~,r)~TFw+E(~,T)(TF—Tw),~
‘
m
V ~
UDF
—
A(~,r)= 11
~DF)
(PDF~WF)
m’O
x [erfc{[(2m + 1) + ~]w2} erfc{[(2m —
.~
~DF)
as
(
1)
+ ~ + KFD~]wF}]
+
1)
—
(~DF~WF)~WF
m0
x [erfc{[(2m + 1) B(~,T)
+
m
V
cTDF — ~
(5.3)
T>O,
—
flw2} erfc{[(2m —
m[erfc{[(2m
~ (~2vw5)
(TDF) naO UDF
+1)
+
~
~]wF}
+
KpD~]WF}]
—
~WF
erfc{[(2m + 1)
—
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
g~,r,—1, ~)+g~,r, 1~i + ÔFD),
~,—1+ôD),
~:1+~FD,
C(~,r) as g~, T,
D(~,r) as ~ ~
r,
ôFW,
0),
359
—1 +
(5.4)
~FD’
+
~~FW’
E(~,T)as ( +~WF~WF)
~ ,n=O
(~FPWF)m[~Ferfc{[2(m+ 1)+
~]wF}
—erfc{(2m
—
~)wF}].
The functions g~( ~, ‘r, a, b) are defined in appendix E (eq. E.26). In these expressions, KFD as is the square root of the ratio of the FL thermal diffusivity to the DP thermal diffusivity, with K as k/pc~, ~ as x/d is the nondimensional distance, d is the thickness of the FL, ~ 6 as \/PeF/4r, 2/KFtexp is the Péclet number of the foil/film layer, rtexp as tItexp is the nondimensional time scale, PeF as d is the time scale of the experiment, ~ as 6 2~/d, and 8D as ÔFD/d. Also, we have ~
2, °~DF as
~WF as (kWPWcPW/kFpFcPF)112,
KF/ KD
(5.5a,b)
(kDpDcpD/kFpFcpF)U
which are the thermal inertia “mismatches” (Carslaw and Jaeger [58],p. 321) between DP and FL, and SW and FL, at the pressure of the FL—SW interface. Also we have ~w 2—(crw2
l)I(u~2+ 1),
PDF_(UDF1)/(UDF+
1).
(5.6a,b)
In (5 .5a,b), k, p and c~,are the thermal conductivity, density and specific heat at constant pressure, respectively, of the designated material for the state of at the pressure of k, thep SW. To 0W6 at high pressure, weeach needmaterial the appropriate values of and c~,. estimate follows the values tTDF and Density fromofthe impedance match and release calculations, while the specific heat at constant pressure results from the classical thermodynamic models discussed above. Assuming that the thermal conductivity, k, may be written in terms of lattice, k~,and electronic, ke, components, i.e. k = k~+ ke, we assume that k ke for metallic target components. In this case, we calculate kD and kF as a function of temperature from the Wiedemann—Franz—Lorenz relation (e.g. Berman [59]) = 2.45 x 10-8 W fl/K2,
(5.7)
where Pc is the electrical resistivity, which we estimate from electrical resistance data on shocked metals, respectively. Assuming the thermal conductivity of the TW material is controlled by lattice processes, we may use the thermal conductivity model of Roufosse and Kiemens [60] to estimate k~. As compared to kD or kF, k~ predicted from this model increases much more slowly with pressure, partially accounting for the development of a significant thermal inertia mismatch across the FL—SW target interface. Based on the release/reshock calculations presented above, we calculate 0~WFfor the Fe—Al 203 and Fe—LiF interfaces using the value of electrical conductivity for 2-Fe given in table 4.1. This value comes from Keeler [39], who summarized electrical conductivity data on 2-Fe between 20 and 140 GPa, and it represents an extrapolation of the trend in the c-Fe data down to standard pressure. We list results of this calculation in table 5.1. As is evident, Fe is more closely matched to Al203 than LiF; since T1(r) is proportional to o~~I(l + o-~~) [see eq. (5.8) below], there is a greater adjustment of T1(r) at the Fe—A1203 interface, as is shown below.
B.
360
Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
lable 5.1 STP and high-pressure thermal-inertia mismatch estimates
STP
100 GPa
200 GPa
ideal interface Fe—Al
203 Fe—LiF
0.56 0.20
0.25 0.11
0.15 0.05
Fe—A1203 Fe—LiF
0.56 0.20
reshocked interface 0.15 0.07 0.08 0.01
We now focus on the FL—SW interface temperature T1(r) as TF(O, r) = T~(0,T), since it is responsible for controlling the interface source radiation intensity. Specializing the relation for T1(r) to the case where the DP and FL are the same material (e.g. Fe), we have TD = TF, °DF = 1, and so ~DF = 0 via eq. (5.6b); we also assume that iXTDF = ~TFD for simplicity. In this context, the same equilibrium thermodynamic state is assumed to exist on either side of the DP—FL interface, and T1(r) is given by T1(r)
G(r)
TF+G(r)ATDF+D(r)~TFw+ l+UWF (T~ TF),
=
1+1
{erfc[(1
—
~D)wF]
—
erfc[(1
+
ô~)w2]},
°WF
(5.8)
D(r) = 1+1 ~WF
erfc(8WwF).
(5.9)
Noting that erfc(x) increases, while erf(x) decreases, as x decreases, we see that D(r) will decay with time, while G(r) can either grow or decay. Most radiation observations constrain a decreasing temperature with time (see discussion below); however, there may be some suggestion of the influence of ATDF on T1(r) in the data on the Fe—LiF interfaces discussed below. If we assume z~TDF= 0 = z~TFDand/or 5~= 0 = eq. (5.8) reduces to T1(r) = TF
+
D(r) z~TFW+ 1
+UWF
(T~ TF), —
(5.10)
which is the reshock model considered by Grover and Urtiew [9]. Further, if we let 6~—+0, we have, from (5.10), TJ(T)-+T!dasTF+
(T~—TF),
(5.11)
relating the temperature of the smooth FL—SW interface, TId, to the temperature 0WF~~ of the ~direct-release Clearly T state, TF. Note that TId = T1(c’D) and that TId approaches TF as ~r~2—*0, and T~as 1, as given by (5.10), will be time dependent only if the FL—SW interface is reshocked. We use (5.10) with the reshock 0D2 stateand temperature in the FLTat the FL—SW interface (TF + ~ TFW), the temperature of the 0wF’ to calculate SW (T~), 1(0), which is labeled “reshock conduction” in figs. 0WF~ 4.5a,b. As Similarly, we use (5.11) to calculate the “release conduction” temperatures from TF, T~and stated above, the Fe—LiF thermal mismatch is greater (i.e., ~ is much smaller, see table 5.1) than that of the Fe—Al 203 interface, mainly because LiF is more compressible and less conductive
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
361
(thermally, see table 4.1) than Al203. In this case, the Fe—LiF interface temperature remains closer to the temperature of Fe at the interface than does the Fe—A1203 interface temperature. Further, the greater compressibility of LiF gives it a much higher compressed temperature than A1203. For example, TH for LiF [via eq. (4.51)] at 160 GPa is ~4200 K (ignoring the possibility of melting), while TH for A12O3 [also via eq. (4.51)] at 230 GPa is ~~c2750 K. The temperature mismatch is much less across the Fe—LiF interface, and the effect of thermal inertia mismatch on T1 is less extreme. In figs. 5.2, 5.3, 5.4 and 5.5, we present calculations for TF(~,T) and ~ r) from (5.3) and (E.27), respectively, and the associated T1(T) = TF(O, T), with ZXTDF = Z~TFD,etc., as assumed to write (5.8), for Fe—Fe—SW targets. To construct these figures, we calculate the compressed/released and reshocked/released states achieved in an Fe—Fe—A12O3 target impacted by a Ta projectile at a velocity _______________________________________________ gi Os Film/Foil Window 0
to300-ns
a -1 ND Distance C C ~ F-
J Pedal ,asam5a,~l.O R.shockad ley.rO,1
000a
Interlace-I t°O
DODD
tinlinity
—
—
C I_C
2 .-‘
I.
‘~-
-
to
~oaoooo*OaoooOOoooaOooooOGOoOaooaaooooOOeOOO0OOOOOO
0. at
c 000000000000D00000000000000000D00000D00000000000000
b I
~. 1
75
150 Time (ns)
225
300
Fig. 5.2. Variation of the temperature near a reshocked Fe film/foil—A1203 interface. (a) The variation of the temperature in the FL, ~ r), and SW, ~ r), as a function of nondimensional (ND) position, ~, with respect to the FL—SW interface (~ = x/d = 0) at four different times during a 300 ns experiment. The nondimensional range of—ito 1 corresponds to —d to d, where d is the thickness of the FL layer. “Reshocked layer” refers to the nondimensional thickness of the reshocked layer, 8~. (b) The corresponding variation of the interface temperature, T1(t).
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
362
Film/Foil
Window
C ~-
g
Initial-eanditians
—
-
‘,
~.
t3-ass
—
..
- -
t30-ns
--
t300-ns
.-..
ar
C 1
-i
0 ND
1
Distance
C 0
I
0,1 N
-
Pactel ,ssamRar=1 0 R5s5ocked leye,’=0.5
—
— ~
DODD
~.
lnterface-T tiflfixs~ty
C C
2°a.’ 10
-
0
~
c
C C
a—
0,
—
—
—
— —
-
00000000000000000000000D0000000000
DODD DO DODODO00000
b o
75
iSO Time (ns)
225
300
Fig. 5.3. Variation of the temperature near a reshocked Fe fihn/foil—Al,03 interface. The reshocked layer depicted in (a) is thicker than that of fig. 5.2, and relative to the conduction length scale (as represented by the Péclet number), causing the temperature of the reshocked layer, and so that of the FL—SW interface, to decay more slowly. In (b), the “t = 0” curve corresponds to T1(0), while that labeled “t = infinity” corresponds to T~(cc). The magnitude of these asymptotic values of T1(t) is governed by that of the FL—SW thermal-inertia mismatch, FrWF.
of 5.67 km / 5; we assume that the calculated reshock temperatures at the DP—FL and FL—SW interfaces are the initial values. This impact velocity is that of one of the experiments (Fe—Fe film—Al203) discussed below. The basic result here is the dependence of the rate of change of T1(r) on PeF and 6~. From (5.8), the change in T1(r) with time is given by dT1(r)/dr
G’(T)~TDF+D’(T)~TFw,
(~) p
G’(r)
1 2 ~WF
D’(r) =
l+tTWF —2
(5.12)
1/2
[(1
-
~FD)
exp{-[(l
-
7TT
4’~TT (PeF)
2]. u26~
exp[—(~wwF)
~D)wF]2}
-
(1
+
6~)exp{-[(l
+ 6*)]2}]
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces I
I
Film/Foil
Window
~~ ~ g
363
-
—
—
- -
Initial-canditians t3-ns
----S.-
.....
-
.....
t30-ns t300-ns
~
a
I 0
C~
-i
1
ND Distance C
I
I
I
N
aneu — —
Rechecked Pactel num.be,=O.1 Leejee=0.5
DODD
I_
t=O Intertece-T tlnfinity
C C 10 0000*ooooooo000a000aoa000000000000000aa000000a0000
a.--
O
DD000Dn0000n000000D000000D000
00000 00000000000000000
C C
0
75
I
150
Time (ns)
I
225
C
3bt’
Fig. 5.4. Variation of the temperature near a reshocked Fe film/foil—A1203 interface. In this figure and fig. 5.5, we hold the reshocked layer thickness constant and vary the Péclet number, or conduction length scale, ~ of the FL. Since the Péclet number is inversely proportional to the conduction length scale, a relatively small Péclet number (0.1) results in a fast decay of T1(t), as shown in (b).
For the particular case we have plotted, and as noted above, unless ~ L ~ 1, and/or i~TD~‘z~TFW,the ~TFw term dominates T1(r). Since the coefficient of the dominating term, D’(T), is always negative, the rate of change of T1(r) will be negative, and T1(r) will consequently decrease with 2),with time. the D’(r) term in (5.12) is dominant, dT1/dT is of proportional to texp~ exp(—~ as ~ Further,_when FWVPeF/4r = ôFw/2\/~. In other words, over the time scale the_experiment ~ represents the ratio of the layer thickness ~FW to the conductive length scale VKFtCXP. Note that ~t exp(— ~2) achieves a maximum value near /L 1 and is much smaller (—0) for 1u much greater or less than unity. In figs. 5.2 and 5.3, we hold PeF constant (i.e. the conductive length scale \/K2t05~) and vary the layer thickness 6~. As shown in fig. 5.2, with PeF = 1, T1(r) for a thin reshocked layer ~ = 0.1) relaxes very quickly (i.e. faster than can be resolved experimentally) to near T1(cx~),while in fig. 5.3, we see that a thicker reshocked layer (~~ = 0.5) will relax much more slowly, and on a resolvable time scale. A similar set of events holds if we fix the layer thickness and vary the conductive length scale of —~
——
364
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces I
-
Film/Foil
Window
--
i—
g
—
—
initial-conditians
- -
ta.3-n, t35-ns
--
t300-ns
.,
—
-
a
0--i______________________________
-l
0 ND Distance
C N Pactel rnsraber—10.0 Renhacked tasjer—0 5
_
0000
interface-I t=ø
DODD
ta.infmity
—
—
C 0
a I..
G-G.eG4G.~Q.Q000.50o00oo0oo0oo000oooooo00oo00ooooo00o
0.
—
-
-
-
-
-
a) Io
C
0 00000000000000000000
D000D 00000000 0000000 DODODO00000
0
0
76
150
225
300
Time (ns)
Fig. 5.5. Variation of the temperature near a reshocked Fe film/foil—Al,0
3 interface. With a Péclet number of 10, the conduction length scale of the FL is small relative to the thickness of the FL, and the interface temperature decays very little over the time scale of the experiment.
the FL, as we show in figs. 5.4 and 5.5. For a conductive length scale large compared to the reshocked-layer thickness (fig. 5.4, PeF = 0.1 and ~ = 0.5), T1(r) relaxes relatively quickly, whereas if the conductive length scale is small relative to the layer thickness (fig. 5.5, PeF = 10 and 6~ = 0.5), 5~P trade off there is little or no resolvable relaxation of T1(T) away from T1(0). Obviously, ~FW and V in their effects on T 1 (T), introducing some ambiguity; only their ratio has a distinct effect on T1 (T). In any event, for PeF 1 and intermediate ~ —0.3—0.7) reshock-layer thicknesses, T1(t) is time dependent on an experimentally resolvable time scale, and its variation with time produces a corresponding radiation source time dependence, as we show in the next section.
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
365
6. Radiative transport in the target With a model of the initial temperature profile of the target components and interfaces, we now establish a connection between the radiation intensity of sources at these temperatures and the radiation intensity emerging from the free surface of the TW during the experiment. The target is represented as a series of plane-parallel layers (fig. 2.2) with optically smooth boundaries (Boslough [8], and appendix F). We assume that (1) source radiation is collimated by the target geometry, (2) all radiation sources are thermal, and so their intensity is given by the Planck function, (3) sources are located only at the FL—SW interface and/or uniformly throughout the SW, particularly along the direction of shock propagation, and (4) all optical properties are independent of wavelength. The model spectral intensity of the radiation emerging from the free surface (FS) of the USW (unshocked window), ‘~mod= IAmod(1k’ t), as a function of wavelength, A, and time after the shock front has passed the FL—SW interface, t, is given by (eq. F.43) 1’Amod(’~, t) as êAsw(t)IAPl(A,
T~)+ êAl(t)IAPI(A, T 1(t))
.
(6.1)
The Hugoniot temperature of the SW, T~,is assumed homogeneous, uniform, and constant, which is consistent with a uniform distribution of radiation sources in the SW. The interface temperature, T1, is a function of time, or constant, in the context of the conduction model discussed above. The subscript A denotes a spectral quantity. In (6.1), we identify ~Asw(~)as ~t’~(t)[1 TASW(t)J[l
+
—
rAITASW(t)]
as ~l~(t)rAsw(t)(1— rAl)
(6.2) (6.3)
as the effectve normal spectral emissivities of the SW and FL—SW interface, respectively. As is evident from (6.1), ~ASW and ~AJ are the properties connecting the intensities of the sources within the target and the intensity emerging from the target. The function ~P,5(t)is defined by as
(1
—
rAFS)TAUSW(t)(1
—
TAsF)
,
(6.4)
and represents the effect on source radiation of propagation through the FS, USW, and SF (shock front). In eqs. (6.2)—(6.4), rAFS, rASF and r~1are the effective normal spectral reflectivities of the FS, SF, and FL—SW interface, respectively. Further, TAUSW(t) as
TASW(t) as
exp[—a~~~~(1 t/texp)1 , —
exp(—a~5~t/t0~~) ,
(6.5) (6.6)
are the effective normal spectral transmissivities of SW and USW layers, respectively. The quantities ~ and a~sware nondimensional forms of the effective normal spectral absorption coefficients in the USW and SW, respectively, and they are given by a~usw
_
aAUSWxFS,
(6.7)
B.
366 I
*
aASW
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
=
.‘
V~\yt~xp)
aASWtSXFS
respectively, where VHW is the material velocity of the SW, and ~ and aASW are the dimensional counterparts of ausw and ~ respectively. Also, XFS is the thickness of the TW, and so the position of the FS with respect to the FL—SW interface (fig. 2.2). Note that ~ and a~uswmediate the explicit time dependence of ~ and ~AI. In writing (6.7) and (6.8), we have also assumed steady shock propagation such that the position of the shock front in the TW (fig. 2.2) may be written xsF(t) as (u
—
v
11~)t(appendix F), where From eqs. (6.7) and (6.8), ~ aASW
(xFs
—
VHWtexp)
VHW
‘
is the material velocity of the SW. and ~ will be of order unity when
a~J~,Z
XFS
‘
(
.
)
which are both —i~~ m~,since the thickness of the TW is generally —i0~m. So for values of aASW and/or aAUSW much larger or smaller than these “geometric” values, source radiation intensity is resolvably or not, affected by propagation through the SW and/or USW, respectively. The USW is usually transparent, so aAUSW 0; if the SW is transparent as well, then aASW 0 and, from (6.2) and (6.3), we have ~AI(~)
~ EAI(O) =
(1
—
TAFs)(1
—
rASF)(l
—
TAI)
,
(6.10)
and ~Asw = 0, respectively. In this case, ‘Amod is governed entirely by sources at the FL—SW interface, and any time dependence of the observed radiation history is due solely to T1. Note that the bound on ~Al in (6.10) is also the initial value of ~AI (i.e., ~Al can only decrease with time). If the TW becomes opaque upon shock compression, we have aASW —+ Again, with aAUSW 0, we have EAI —30 and ~.
EAsw(t) <(1
—
TAFS)(1
—
rASF)
(6.11)
.
In this case, observable sources are confined to the shock front (this is the “ideal case” of Boslough [8]). The impact of these and other model parameters on IAmod(A, t) is more explicitly depicted by writing the partial derivatives of IAOsOd(A, t) with respect to A and t. From eqs. (6.1), (6.2), and (6.3), these are A(l9IAmOd/t3A)t = texp(ôIAmod/19t)A
T~)+ P(~)êAI(t)IApl(A,T1(t))
~ IAmoda~Usw+ {[1
—
+ P(pII)êAI(t)IApl(A,
r~5+ 2rAITASW(t)]IAPI(A, T~)
—
T1(t)) dIn T1/dt
,
—
SIAmod(A,
êAI(t)IApl(A,
t)
,
(6.12)
T1(t))}a~~
(6.13)
P(~)as~t/(1—e~), ,t~~asC2IAT~, ~asC2/AT1. Relation (6.12) exemplifies the fact that the wavelength dependence of ‘Amod is due solely to that of the Planck function, since we have assumed that the optical properties are independent of wavelength. We make this assumption because it is not clear at this point that existing data can resolve wavelength dependent optical properties (Svendsen and Ahrens [61]). Again, for most TWs, we have aAUSW —0. If, in addition, T1(t) is approximately constant with time,
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
367
which may occur in a thick (PeF ~‘ 1) or thin (PeF ~ 1) FL with a thick (6~ 1) or thin (5~ °~1) reshocked layer of FL material, or at a smooth interface, as discussed above, (6.13) reduces to ——
texp(t9lAmod/8t)A
=
{[1
—
2rAI’rASW(t)]IAPI(A,
TAI +
T~)
—
EAI(t)IApl(A,
T 1(t))}a~~~ .
This will be positive if 2rAITASW(t)]IAPI(A, T~)>êAI(t)IAPI(A, T [1 r~1+ 1(t)) ,
(6.14)
(6.15)
—
but otherwise negative, since a~swis always Consequently, finite andofa TAmodpositive. will grow or decay withwith timea on the value basis of of ~the sign time independent interface temperature, ‘ (6.14). If the TW is initially transparent and remains relatively transparent upon compression, we have ~ and ~ Putting these into (6.13), we have texp(ôIAmod/t9t)A
=
P(p~J)ê~ 1I~~1(A, T1(t)) d ln T1/dt
,
(6.16)
and any variation of ~ with time should reflect that of T1 through the Planck function. In particular, ‘Amod will increase or decrease as T1 increases or decreases at a given wavelength. 7. Models and data We compare models and data in the context of the standard al. [63]). In our case, it is given by N~
N 1
as
2 statistic (e.g. Bevington [62],Press et
x
1 —~
i-i j=1 ~
2. [~A~X~(A~, ~)
—
IACOd(A~,t1
a)]
(7.1)
In this relation, 1~exp(Ai,ti), 1Amod(Ai, t 1 a), ~ as o-(A1, t1) are, respectively, the experimental and model spectral radiances, and the experimental uncertainties, all at a particular wavelength, A1, and time, t1. Also, NA and N~are the number of wavelengths and times sampled, respectively, in the experiment. The five-component “vector” a is the model parameter vector, in our case given by
a as {rASF, ~
T~,r~1,T1(t)}
.
(7.2)
Since the radiation model is nonlinear with respect to ~ a~sw, T~and T1(t), (7.1) is not strictly a maximum likelihood measure, even if the data errors are normally distributed. However, if the best-fit values of the a (i.e. amjn) have uncertainties sufficiently small such that the value of ‘Amod can2(a) be well will approximated thethe first two termslikelihood of its Taylor series(Press representation about IAmod(A, t; amin), ~ be “very close”byto maximum estimate et al. [63]). We note that TAFS and ~ are not included in (7.1), since they may be calculated or determined from index of refraction and absorption data for the TW. From the conduction model, we have explicit expressions [e.g. eq. (5.8)] for the time dependence of T 1, which allow us, in principle, to constrain 0~WF’etc., given fitted values of T 1. Similarly, the fitted reflectivities allow us to constrain changes in the indices of refraction across boundaries (e.g. the shock front). Note that, in general, the optical
368
B. Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
properties constrained from (7.1) will not be A-dependent unless we give them, a priori, an explicit A-dependence, with constants whose values are chosen by the fit (i.e. by the data). Since we have no reasonable expectation for this A-dependence, we cannot truly constrain it. It is for this reason, plus the limited resolving power of the data itself (Boslough [10], Svendsen and Ahrens [61]), that we assume a~~\\T, aAUSW, etc., are independent of A in the previous section. However, we may determine an apparent A-dependence of the optical properties if we specialize (7.1) and fit at each wavelength over time, i.e.,
I
I 000a) 450-nm DO~ 600-nm t2~) 750-nm .aSA~900-nm
to l.a
o%
E
0
0
a
0
-
r
II
0
0
260
130
390
Time (ns) c_i Wien —
b
GS
— GSel --LM
40
200
120
20
Time (ns) Fig. 7.1. (a) Observed radiation history and (b)
x2
statistic of greybody model (ê
15, Tg0) applied to radiation data from Fe—Fe film—Al203 target impacted by a tantalum projectile at 5.67 km/s, resulting in an Fe Hugoniot pressure of 244 GPa and an A1203 Hugoniot pressure of 190 GPa, which 2. (c, Greybodyatmodel is also the Fe—A1203 interface pressure (Bass et al. 161). Part (a) displays the radiation intensity (spectral radiance) data,d) collected four wavelengths: 450, 600, 750 nm. Part “goodness” the greybody fit, emissivity, as indicatedé~by Chi2 = x parameters constrained by and data 900 displayed in (b) (a); displays (c) the the best-fit normal of greybody effective 0(t);(d) the corresponding greybody temperature, T10(t). Fits using Wien’s law, Golden Search (GS), GS with the effective emissivity set to 1 (GSeI), and the Levenburg—Marquardt algorithm (LM) are indicated.
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
369
1 ~ [IACXP(AI, t x2(A~a) as j=l ff~ 1)
2 —
jAmod(Ai, t1
(7.3)
.
a)]
Svendsen and Ahrens [61] constrained akmjfl in this manner for radiation data from Ta—Ag—MgO targets. We use a very simple version of this approach below with the data of Bass et al. [6] to constrain aASW.
First, however, it is instructive to consider fits to data using simpler models than that represented by
(7.1) and (7.3). Most earlier workers (e.g. Kormer [1], Urtiew [2], Lyzenga [5], Lyzenga et al. [11]) constrained model parameters via the greybody relation iAgb(A, Egb~ Tgb) as EgbJAPl(A, Tgb)
The associated
x2
(7.4)
.
statistic is given by
Wien —
C
GS
--LM
40
120
200
28u
Time (ns) c
cj
I
I Wien
~ 40
1~0 200 Time (ns) Fig. 7.1. (Cont.)
d
2~u
370
B.
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
1
NA
Xgb(tj; Egb~Tgb) =
i=l
(7.5)
t 2. [1~exp(Ai, 1) JAgb(Ai, tj; Egb~ Tgb)]
~
—
Since the summation in (7.5) averages ~g 5over all observed wavelengths, it represents a wavelengthaveraged effective (i.e. total) emissivity. Given that the only A-dependence in the greybody model is contained in the Planck function, IAPI(A, T), the more closely the data follow the blackbody wavelength distribution at a given temperature, the better the fit [i.e. the lower the value of x~b(tf)].Since both the data and model depend explicitly on A, the fit proceeds over all observed wavelengths at a given time during the radiation history. As a result, ê55 and Tgb are functions of time. Since ‘Agb depends nonlinearly on Tgb, we must find the best fit values of and iteratively with the minimization constraints on To obtain starting values of ~gband Tgb for the nonlinear fit, and for comparison, we may use Wien’s approximation to IAPI(A, T) in X~b(~)’ which follows from IApl(A, T) in the limit exp(C2/AT) ~‘ 1, i.e., ‘Agb
~5.
I
I 0000
0
450-nm
‘~
0
(0
130
260
390
Time (ns)
o_i I
I
C’2
Wien —
GS
GSeI --LM -
~
b ci
40
I
I
120
200
280
Time (ns) 2 statistic for an Fe—Fe foil—LiF target impacted by a tantalum projectile traveling at 5.41 kmls, Fig. 7.2. (a) resulting in an Observed Fe foilradiation Hugoniothistory pressure and of (b) 227 x GPa and an LiF Hugoniot pressure of 122 GPa. (c) Greybody effective emissivity and (d) temperature constrained by radiation data shown in (a).
B. I
371
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
I 5t’ rWgb, — P 1 1i P \ ~ WgbJ‘t = — rWgb1AWi~/t, ~ Wgb)
‘AWgbY
— —
EWgb
1 e ~C~/ATwgt,
—,~—
The relative error incurred in approximating ‘APt by ‘AWi is equal to exp(— C 2/A T); this approximation is accurate- to within 1% for AT <3 x i0~mK (Siegel and Howell [15]). Since we can fit Wien’s relation to the data in a linear least-squares sense, we can solve for ~wgb and Twgb directly (i.e. without iteration). With these values, we may safely apply an iterative technique to (7.5) to constrain and Tgb and be assured of a nondivergent fit. We use the Golden Search (GS) and Levenberg—Marquardt (LM) iterative techniques (Press et al. [63]) to obtain three different fits: (1) GS with êg5 variable, (2) GS with Egb = 1, and (3) LM with ~gb variable. We present a greybody fit to the radiation observations from two experiments of Bass et al. [6] in figs. 7.1 and 7.2. Figure 7.1 displays a fit to data from an experiment on an Fe—Fe film—A1203 target impacted by a Ta projectile traveling at 5.67 The trend in x~b(t)suggests that thev isfitthe getsnumber better 2 ~‘ ±2Vi as km/s. the number of data get very large, where with time. Strictly speaking, x ~gb
——
I
1
0
....
—
40
\
Wien
—GS ——LM
120
200
2~
Time (ns) c ___________________________________
I
I
Wien GS GSe1 --LM —
_
a)
iT:~ITrT~’ c O
01 ~
40
dC
I 120
200
Time (ns) Fig. 7.2. (Cont.)
2~
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
372
of degrees of freedom in the fit (i.e. the number of data minus the number of parameters, 2 in this case); we might hope that X~b —2 represents a reasonable fit for the greybody model. All of the fits show Tgb(t) decreasing with time, and for the variable emissivity fits, êSb(t) increases slightly with time. This behavior is characteristic of most Fe—Fe—A1203 experiments of Bass et al. [6]. For all the Fe experiments, we note that T~~ T1(t), and that the behavior of case, from (6.1), we have jAmod(A,
t)
ê~b(t) is
consistent with assw —0. In this
EAJ(t)IAPI(A, T1(t)) ,
(7.7)
and the decrease of Tgb(t) with time (fig. 7.ld) can be explained in terms of T1(t), as detailed above. Also, the slight increase of ~g5(t)with time (fig. 7.lc) can be explained most simply by a slight decrease of the Al203 absorption coefficient upon compression. This may be consistent1with the observation that between 0.1 and 1 GPa, the refractive index of A1203 seems to decrease with pressure (——0.001 GPa Davis and Vedam [64]). Since aAUSW 0 for Al 203, this observation implies that aASW—O as well. In this case, (7.7) implies that jAmod(A, t)
êAl(0)IAP](A, T1(t)) = (1
—
rAFS)(l
—
rASF)(1
—
rAl)IAPI(A, T1(t))
(7.8)
for the Fe—Fe—A12O3 experiments. In contrast with this last fit, the experimental and greybody fit results displayed in figs. 7.2a—d, for an Fe—Fe—LiF target impacted by a Ta projectile traveling at 5.41 km/s, exhibit a relatively constant greybody temperature (fig. 7.2d) with time and a systematically decaying greybody effective emissivity with time (fig. 7.2c). In this case, Tgb(t) implies a relatively constant T1(t), as we expect for~ a2V~(Ftexp smooth 5FW interface (Grover and Urtiew [9], and eq. (5.11) above) or a reshocked interface with (fig. 5.3b) or 3FW ‘~ 2VKFtCXP (fig. 5.2b). The reshocked interface with ~FW ~° 2’S/KFtexp is less likely than the latter possibility, as implied by the data—model comparison in fig. 4.5b. The behavior of Egb(t) ê~(t)reflects a shock-induced increase in the absorption coefficient (i.e. ~ > aAUSW) of LiF via (6.6), (6.8) and (6.14). Wise and Chhabildas [16] found, via laser interferometry, that LiF remains essentially transparent up to 160 GPa. The fluctuations in the fit after about 160 ns may be due to wave reverberations or other dynamic effects, which are beyond the scope of our model, and/or possibly to the influence of reshock at the Fe—Fe foil interface, as mentioned above. In judging the value of any fit, the resolving power of the data is an issue. In particular, the ability of the data to constrain model parameters may be judged through confidence limits (e.g. Press et al. [63]). We display these for Fe—Fe—A1 2O3 and Fe—Fe—LiF experiments in figs. 7.3 and 7.4, respectively. Parts (a) and (b) represent confidence limits for x~b(50ns) and x~b(250ns), respectively. The shaded (central) region in each diagram represents that part of model parameter space which explains 68.3% of the data. Similarly, the 95.4% and 99.99% regions explain corresponding percentages of the four-wavelength data. Note that these limits are consistent with the trends in x~b(t)(figs. 7.lb and 7.2b). The basic information conveyed by these diagrams is a measure of the uncertainty of the fits; for example, from fig. 7.3, the fitted value of Tgb(SO ns) has an uncertainty of about ±400K at the 68.3% level, and about ±600K at the 95.4% level. As stated above, the interface contribution to the observed intensity dominates the SW contribution (Bass et a!. [6]). On the basis of this observation, we may reasonably fit a simplified version of the full radiation model to the data via eq. (7.5). We do this for the Fe—LiF data fit to the greybody model in fig. 7.5. First, we note that, at t = 0, ‘Amod is, from eq. (6.1),
373
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
6)
-
~Ir
3900
(‘2
5600 7300 Temperature
aL
~95.4%
.~co aa, C ii) 6) 0 Co
9000
bL I
I
68.3%
(0
96.4%
0)
(‘2
Si 5, 4) a) C) .2~
-
5400
I
683% U ~
Co ~9999%-
(‘2
I I 4500 5400 Temperature
1 3300
630u
êAI(0)IAPI(A, T1(0)) = (1
—
rAFS)
4300Temperature 5300
6300
Fig. 7.4. Confidence limits for the LM fit displayed in figs. 7.2c,d near the beginning (50 ns) and end (250 ns) of radiation histories fit by greybody model.
Fig. 7.3. Confidence limits for the LM fit displayed in figs. 7.lc,d near the beginning (50 ns) and end (250 ns) of that part of the radiation histories fit by the model. The designations 68.3%, 95.4% and 99.99% refer to that fraction of the data (which total 4) satisfied by the range of model values (~ 8b’ Tlb) encompassed within the appropriate regions and contours, subject to the assumption that the experimental errors are normally distributed.
1Amod(’~,0)
66.3% 954%
C
~9_~
3600
I I I I 3800Temperature 4600
-
3000
e°~’-’~”(1 rASF)(1 —
—
rAI)IAPI(A,
T1(0))
.
(7.9)
In this case, the magnitude of ~ A, 0) is controlled by the reflectivities, ~ and the initial value 0WF through eq. (5.10) in the simplest of T1, The which is dependent values of TF, T~and case. greybody fits inonfig.the7.2 suggest that,~TFW, for this experiment at least, T 1(t) is approximately constant. Assuming this and ~ = 0, we approximate eq. (6.1) as ~~mod(A, t) =
~AJ(t)IApl(A,T1(0)) = (1
—
rAFS)(l
—
rASF)(l
—
rAt) ea~6wh/tc0pIAPl’~ ‘A,
T1(0)).
(7.10)
In this case, the time dependence of ~ is due solely to the SW transmissivity. Using (7.10) in (7.3), along with T1(0) = Tgb(O), we may fit the Fe—LiF data for ~ We present the results of this fit in fig. 7.5. The data are cut off at 160 ns to reduce the influence of possible dynamics or Fe—Fe interface reshock on the fit. The parameter values resulting from this fit are given in table 7.1. We eliminate TAFS
B.
374
Svend.sen et al., Optical radiation from shock-compressed materials and interfaces
E~_i I
I 0000 450-nm
0000 600-nm
i
<~~ri-~
::00ooooEo~~:
40
80
120
160
Time (ns) Fig. 7.5. Time dependent spectral radiance fits to the data displayed in fig. 7.3a for the Fe—Fe foil—LiF target. We fit eq. (7.10) to the first half of the Fe—LiF data displayed in fig. 7.3a via eq. (7.3). The slope of each continuous curve, representing the fit for the corresponding wavelength, constrains the effective normal absorption coefficient of the shocked LiF, while the intercept constrains the Fe—LiF interface and shock front effective normal reflectivities.
Table 7.1 Simplified radiation model parameters. For this fit, ~ = 4.15 mm, = 390 ns, T 1 = 4200K and rAFS = 0.08 at all wavelengths wavelength (nm)
(1— r1~~)(l — r)1)
(m’)
450
0.76
137
600 750 900
0.56 0.68 0.67
134 125 122
aASW
from the fit since it is equal to 0.08 for LiF, as estimated from n = 1.39, the index of refraction of LiF 1at STP (Weast [37]). Since thewe index of refraction seems to increase (——0.002 GPa Burnstein and Smith [65]), expect rASF ~ rAFSofforLiF LiF. Clearly, LiF haswith lost pressure some transparency upon compression. The trend in aASW toward lower values at longer wavelengths is unresolved but consistent with the Bouguer’s law expectation that a,~= 41T~~e~’/k, if we, the electromagnetic extinction coefficient (Siegel and Howell [15], p. 427) is constant or varies inversely with A. As suggested above, similar fits for Fe—Al 203 imply that aAUSW > aASW, an intriguing possibility which we do not yet understand. Lastly, we take the results of the greybody fit shown in fig. 7. id for the Fe—Fe—Al203 experiment, assume Tgb(t) = T1(t), and use (5.10) to write 1~
~ [Tgb(O)
—
Tgb(texp)].
(7.11)
~TFw = erfC(~ With Tgb(O) Tgb(texp) = 1200 K from the fit displayed in fig. 7.ld, we may calculate the trade-off between the FL—SW interface temperature due to reshock, ~TFW, and the ratio of the reshocked layer —
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
C C C
_____________________________________________________
—
—
—
—
—
—
1
—
- --
—
—
C
375
1
0.5
I
1.0
1.5
0_\VFO-
-
~i_2.0 2.0
6~/~ Fig. 7.6. Magnitude of temperature at FL—sw interface due to reshock, ~ versus the ratio of the reshocked layer thickness, ~ to the conduction length scale, ‘/~. This trade-off is constrained by the magnitude of Tgb(O) — T 65(ç5) from the greybody fit for the Fe—Fe—Al20, data displayed in fig. 7.2a. 8FW’ to the FL conduction length scale, VKFtCXP, for different values of the FL—SW interface thickness, thermal-inertia mismatch, 0~wF~These calculations, displayed in fig. 7.6, imply that the larger the reshocked layer thickness relative to the FL conduction length scale, the higher the reshock temperature at a given thermal mismatch. For this particular experiment, we expect 0~WF~ 0.1 from calculations discussed above; we also expect I.~TFws 2000 K from the calculations presented in fig. 4.5a. In this case, fig. 7.6 and model calculations imply that ~FW 2VKFtexp i0~m. Since this is a film experiment, with d — 10_6 m, we tentatively conclude that all of the film layer experienced reshock in this experiment. ~
——
8. Summary In this work we have attempted to construct a thermomechanical model for a shocked target that can be used to explain and interpret shock-induced radiation observed from such a target. We have emphasized (1) the potential effects of compression, release/reshock, and equilibrium phase transitions on the “initial” temperature profile in the target, (2) the effect of conduction on this temperature profile over the time scale of the experiment, and (3) the effect of propagation through the dynamic target geometry on radiation source intensity and its temporal variation. Comparison of the model with the results of experiments on Fe—Fe—LiF and Fe—Fe--Al 203 targets suggests the following: (1) Release / reshock calculations for Fe—Fe—Al203 targets, in comparison with the experimental results of Bass et al. [6], suggest that Fe experiences approximately 200—1500 K of reshock heating at
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B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
both Fe foil—Al203 and Fe film—A1203 interfaces when released from as245—300 GPa to interface pressures of 190—230 GPa. Below 190 GPa, reshock for Fe—A1203 interfaces appears to be minimal. Both the data and calculations suggest that the degree of reshock is strongly pressure dependent, which is consistent with the results of Urtiew and Grover [7]. In contrast, Fe released from the same range of Hugoniot pressures to Fe—LiF interface pressures between =130 and 160 GPa experiences little or no reshock. This more ideal nature of Fe—LiF interfaces is enhanced by the fact that, besides being a poorer shock impedance match to Fe than Al203, it is also a poorer thermal match, resulting in less change in the interface temperature away from the Fe release-state temperature. Comparison of data and calculations for both of these windows suggest that, while attention to the initial conditions of the interface is essential to minimize reshock, a more important factor may be the choice of window. (2) In the absence of energy sources and significant energy flux from other parts of the target, ~the 2),where = rate of change of the interface temperature,_T1(t), is proportional to exp(—~t 6Fw/2V~. For Fe at FL—SW interfaces, ~ 10 urn; consequently, a 100 urn reshocked Fe layer would relax very little, remaining near T 1(0) on the time scale of the experiment. However, if 8FW 1 um, T 6FW 1(t) relaxes almost instantaneously to its value T1(oo). T1(t) is resolvably time dependent for(3) Greybody fits to an Fe film—A1 203 experiment of Bass et al. [6] show the greybody effective emissivity, Egb (t), to increase slightly with time, while the greybody temperature, Tgb (t), decreases with time. This behavior is characteristic of most Fe—Al203 experiments. The decrease can be 8FW 2V~KFtexp for of thisTgb(t) experiment. explained in terms of the model for T1(t), and it implies that Further, assuming Tgb(t) = T the~ greybodyA fit constrains theofamount 0WF —0.1 and1(t), 8FW ~ slight decrease the A1 of reshock, ~TFw, to be s2000 K with 203 absorption coefficient upon compression can explain the slight increase of êg5(t) with time. This may be consistent with the observation that the refractive index of A12O3 seems to decrease with pressure. In contrast, greybody fits to data from an Fe—Fe foil—L1F target show a relatively constant greybody temperature and decreasing greybody emissivity. The constant greybody temperature implies a constant interface temperature, as we expect for an interface experiencing minimal reshock, while the decaying is consistent with a shock-induced increase in the absorption coefficient of LiF. Setting T1(0) = Tgb(O), we 1 (table 7.1) for fit a simplified of in thethis fullexperiment. radiation model to these data to find aASW 100 m LiF, shocked toversion 122 GPa (4) Finally, we note that the equilibrium thermodynamic estimate of the Hugoniot temperature of Fe is strongly influenced by electronic and/or anharmonic contributions to c~,at high pressure, as evidenced by both (1) the results of Boness et al. [45] when used in eqs. (4.46), (4.50) or (4.53), and (2) by requiring the solid-Fe Hugoniot and an extrapolation of the experimentally constrained Fe melting curve (Williams and Jeanloz [50]) referenced to this Hugoniot, to intersect at 245 GPa (Brown and McQueen [44]). This last constraint provides a value of ~Q(p 2, as compared to 1) = 0.046 J/kg K F( p,) = 0.090 J / kg K2 from the work of Boness et al. [45], suggesting some anharmonic contribution to c~of e-Fe. These results substantiate the arguments of Brown and McQueen [44, 22] for the importance of including electronic contributions to c~when calculating T~of shock-compressed metals. —
—~
——
—
~gb(t)
‘—
Acknowledgements We thank William W. Anderson, Mark B. Boslough, A. James Friedson, Hua Tan, Dion L. Heinz, Douglas R. Schmitt, and James A. Tyburczy for enlightening discussions. We also thank J. Michael
B. Svenolsen et al., Optical radiation from shock-compressed materials and interfaces
377
Brown and two anonymous reviewers for constructive comments on an earlier version of this work. Support from NSF grants EAR-8608249 and 8608969 is gratefully acknowledged. Contribution 4488, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125.
Appendix A. Shock front and target boundary conditions In the text we have attempted to detail the influence of certain processes, such as release and reshock, conductive and radiative transport, etc., on observed shock-induced radiation. Although microscopic processes lie at the heart of shock-induced radiation, the experimental foundation of our work requires an interpretive, conceptual framework much more abstract and general than any particular microscopic physical theory can provide. This framework is provided by continuum mechanics, which in principle enfolds and abstracts all microscopic models and viewpoints, via their appropriate macroscopic (e.g. thermodynamic) limit. With this framework, we have a representation of the phenomena partial to no particular “microphysics”, and so accessible to all. The purpose of this part of the appendix is to outline and detail the continuum framework we use to represent and interpret experimental results on shock-compressed materials. A. 1. Boundary conditions across a moving, nonmaterial surface As discussed in the text, the target is usually composed of two or three different materials sandwiched together in a plane-layer-style geometry (see fig. 2.1). The shock wave propagating through this target is a three-dimensional region with some thickness 6 (typically ~10_8 m in the materials of interest here, Kormer [1]), and moving through the material with a velocity of propagation u (typically -— io~—io~ mis). Shock compression produces large (factor of 2 to orders of magnitude) changes in any given local thermomechanical (TM) field (density) ç& over the length scale 6 and on a very short time scale (i.e., 6/~u! 10h1_1012 s), resulting in large gradients (—I~’I/6) in these TM fields across the shock front region. We assume that these length and time scales are sufficiently short so that, from the macroscopic viewpoint, they may be idealized as infinitesimal and instantaneous, respectively. In this case, we may idealize the shock front macroscopically as a moving surface S. Let the material through which S propagates occupy a spatial volume V in Euclidean space. Then S divides V into regions “ahead” (Vt) and “behind” (V) the shock front. This surface representation for the shock front is defined by the limit 6 —*0, and we note that in this limit the gradients of ~i across the shock front become infinite. Consequently, cia- loses a continuous representation in V, becoming discontinuous across S in this representation. In this case, S is referred to as singular (Truesdell and Toupin [66], section 173) with respect to ~fr,such that ——
[]as~i—~$O,
(A.1)
where [Ici’I] is the “jump” of cli across S, and ~ and cli are the limiting values of “just” ahead and behind S, respectively. In the text, we designate the states behind (—) and ahead (+) of the shock front in a more conventional fashion via the subscripts “H” and “i”, respectively. Note that S is oriented such that U(~)as u~b >0, where 1’ is the unit normal to S. The concept of a singular surface forms the basis for the continuum mechanical description of shock compression. For our purposes, we assume that S ,/j
B.
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possesses no fields or properties other than a motion (i.e. u) independent of the material on either side of it in V. Assuming further that all kinematic and TM fields defined on S are homogeneous there, the
balances of mass, linear momentum, energy and entropy across S are given by [66, sections 193, 205, 241, 258] [~pU(~)j]=0,
(A.2)
[~pU(~)vj]+ [~t~]=0,
(A.3)
[IpU(~)(e + ~
v) + (v . t
—
q(~))~] as 0,
(A.4)
[lpU(~)s— T’q(~]0.
(A.5)
In (A.2)—(A.5), U~) U . 1’ is the component of the propagation velocity u as u v~of S normal to S, t o• b is the Cauchy stress vector, o~is the Cauchy stress tensor, q(~) q 1’ is the component of the heat flux normal to 5, q is the heat flux vector, e is the specific internal energy, s is the specific entropy, and T is the absolute temperature. Note that the entropy balance is an inequality via the requirement of a nonnegative entropy production on the shock front. The mass balance, eq. (A.2), is the first of the relations we use to interpret experimental results. Using (A.2) in (A.3)—(A.5), we obtain ±
p~U~)[~v~] + [~t~] = 0, + ~v.vj] +[~vt—q(~]=0, 1q(~]0 , p~U~1[~s~] [~T
—
(A.6) (A.7)
(A.8)
—
respectively. Note that (A.6)—(A.8) represent five equations in 24 unknowns: ~ U~,v~,t~,e~,q~), s~and T~. From the physical viewpoint, shock compression produces a “sudden” change in the velocity of the material in a direction normal to the shock front. In the context of the singular surface representation of the shock front, we idealize this change as a discontinuity in the component of the material velocity normal to S, i.e., [~v(~]0,
(A.9)
where 13(p) v~b. Also, note that [ju~]= 0. To substitute (A.9) into (A.6)—(A.8), we must first cast these latter relations into normal and tangential forms. To do this, we define ‘T 0 (a = 1, 2) as vectors tangent to S such that V_TIXT2I~TIXT2!,
i.e.,
~
normal
(A.10)
r~and r2 form a right-handed system. With these, we may resolve any vector into components and tangential to S. In particular, we have
V=V(V)V+V~)Ta
,
(All)
B. t_ t(j~)V+
t(T)Ta
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
379
(A.12)
.
Noting that i’~~=l,
(A.13)
V~T=O,
and assuming S is planar, that
10,
a~/3,
(A.l4) we may substitute (A.12) and (A.13) into (A.6)—(A.8) to obtain
p~U~)[Iv(~)I] + [~t(~)J] =0 PU(.?)[IV(T)I]
+
[Jt~)~] = 0
pU(~)[Ie+ ~(v~) PU(I,)[ISI]
+
,
(A.15)
,
(A.16)
v(~)~v~))I] + [~u(~)t(~) + V(~)~t~) q(~)I]= 0, —
[lT~q(~)I] 0.
—
(A.l7) (A.18)
To proceed further, we make the constitutive assumption that the material on either side of the shock front is an ideal (also known as barotropic or elastic) fluid, i.e., o——P~I.
(A.19)
With this, we have t(~)=
=
jj~
as
—P~i’•b = —P~
(A.20)
via (A.13), and = O.±.
= J/~
=
p~~’
=
0
(A.21)
from (A.14). Putting (A.20) and (A.21) into (A.15)—(A.18), we obtain
[~PI] as p~U~)[Jv(~)J],
(A.22)
p~U~)[Jv~]= 0,
(A.23)
p~U~)[~e + ~V~)I] + [~v(~)t(~)
—
PU(j,)[I5J]
—
[IT~q(~)I] 0,
q(~)I]= 0,
(A.24) (A.25)
where we have used (A.23) in (A.17) to obtain (A.24). Relation (A.22) is the second of the relations
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Svendsen et al., Optical radiation from shock-compressed materials and interfaces
used to interpret experimental results. From the momentum balance, eq. (A. 15), we may write p ~U~)[~ ~v~]
+ [~v(~)t(b)~] = (v~t~ V(~)t(~)) + ~[~v(~)t(~]. ~
—
(A.26)
Putting this into the energy relation, eq. (A.24), we obtain p~U~)[~e~] = ~(v~)t)
V~)t~))
-
-
~[~v(~)t(~]
+
[~q(~]
-
~[~v(~](t~) + t~))+ [~q(~].
(A.27)
From the mass balance, we have =
~
[jp~]U~
(A.28)
.
Putting this into (A.27), we obtain
[~e~]= ~
~±
[~p~](t~ + t~))+
[~q(~]
=
-~[~1Ip~](P+ P~)+ PU~) [~q(~],
(A.29) where we have used the ideal fluid constitutive assumption (A. 19). Finally, if we make the constitutive assumption that the shock front is adiabatic, i.e., [~q(~] = 0, (A.29) reduces to ~(P
[~e~]=
+
Pt),
(A.30)
with ‘q 1 p ~ - being the relative compression. Relation (A.30), known as the Rankine—Hugoniot relation (e.g. Rice et al. [20]), is the third relation used to interpret experimental shock compression data. With the adiabaticity assumption, (A.25) reduces to —
p~U~)[~s~] [~1/T~]q~) .
(A.31)
If we further assume no heat flow, i.e., q~)= 0, (A.31) reduces to
[Isl]
0,
(A.32)
implying that the specific entropy must increase across an adiabatic shock front separating two non-heat-conducting, ideal fluids, as previously established by other means (e.g. Bethe [67]). Relations (A.2), (A.22) and (A.30) represent the balances of mass, momentum and energy, respectively, used in this work to interpret experimental shock compression results, and for various related calculations of Hugoniot, or shock-compressed, states. Along with (A.32), they constitute a thermodynamic description for the “experimental” shock front.
A.2. Target boundary conditions The shock front represents a kind of boundary in the target, across which we may use, given the necessary experimental or other information, (A .2), (A.22) and (A.30) to calculate the change in
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
381
density, material velocity, pressure, specific internal energy, etc., during shock compression. As stated above, however, different materials make up the target, introducing a further “discontinuity” into the field description of the target as a whole. Consequently, we must also find the balance of mass, momentum, energy and entropy across the boundaries between target layers. Since we assume that, as each target layer is compressed and the shock front passes on into the next target layer during the experiment, the layers do not separate or blow apart at their interface, this boundary may be regarded as a material surface, such that U(b) = V~), or U~1= 0. This states that, at a material surface or interface, the surface moves with the displacement velocity of the material on either side of it. Consequently, at such an interface [Jv(~] = 0, and the balance of momentum becomes [~t(~)~]=0,
[~t~,.)~]=0,
(A.33)
from (A.15) and (A.16), respectively. Similarly, the balance of energy, eq. (A.17), reduces to [~V(~.)a, ]t
= [~q(i,)
],
(A.34)
where we have used (A.33) in (A. 17) as well. Under these circumstances, the entropy inequality (A. 18) simplifies to [~T~q~]~
0.
(A.35)
Substituting the constitutive assumption (A. 19) into (A.33), the balance of momentum at a material interface in the target takes the form [~P~]= 0.
(A.36)
Under the constitutive assumption (A.19), we also see from (A.21) that t~ are identically zero, and with this the balance of energy (A.34) then requires that the normal component of the heat flux be continuous across the interface, i.e., [~q(~]
=
0.
(A.37)
Substituting (A.37) into the entropy balance (A.35), we have q(~)[ITU0, implying that the interface between the two ideal fluids will achieve thermal equilibrium (i.e. no entropy production) if (1) either ideal fluid (and so both) is adiabatic (q~)as 0) and/or (2) temperature is continuous across the interface, i.e., [I T~]=
0.
(A.38)
We assume (A.38) in all energy transport models presented in this work (see appendices D and E). Relations (A.36)—(A.38) then represent boundary conditions appropriate for the assumed constitutive nature of each target component, and consequently are consistent with the analogous shock front relations given above.
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et al., Optical radiation from shock-compressed materials and interfaces
A.3. Shock velocity—material velocity relation
Beyond the basic assumptions discussed in part A.1 underlying the balances of mass (A.2), momentum (A.22), and energy (A.30), the basis of the relations between the Hugoniot and isentropic properties is the constitutive assumption that the shock front propagation velocity U1 may be written as
[~vI] as
a function of as
u
=
—
v~in a Taylor series about v1. To second order, this is
—
+2o([~v~]~). a1 + b1[~v~]+ c~[~v~]
(A.39)
The coefficients in (A.39) are defined by a~ U~~
11511..0,
(A.40)
b~as dU1/d[~v~]~[1U1]..0,
(A.41)
c~
~
~
(A.42)
.
Now, using the relative compression, ~ = 1 form
—
P1/PH’
as
defined in the text, we may write (A.2) in the
[IvI]=,1U1.
(A.43)
Following Prieto and Renero [68], we now substitute (A.43) into (A.39), rearrange, and square both sides of the result to obtain 4U~[(1 b 2 2a~c 2]U~ + a~= 0, (A.44) c~rj 1?)) 11) i.e., a biquadratic relation for U 1 in(A.44) terms has of a1, c1 and As shown by Prieto and Renero [68],for 2/4a?)2, theb1,solution b1’q <1 and c1 (1 b,fl) U~= a~/{(l b 2 2a 2}. (A.45) 1rj) 1c1?) —
—
—
~.
~
—
—
—
Substituting (A.43) into (A.22), we have either U~= (1/p~?))[~P~],
(A.46)
or [IvI]2= (?)ip
1)[~P~],
(A.47)
depending on whether or not we eliminate [~v] or U1, respectively. Putting (A.45) into (A.46), we obtain 2—2a 2} (A.48) [~PI] = p~a~~/{(1 b1~) 1c~~ Note that c 1, when resolvable by experimental data, is usually negative (e.g. Pastine and Piacesi [69], .
—
B. Svendsen a a!., Optical radiation from shock-compressed materials and interfaces
383
Ruoff [40], Prieto and Renero [68],Brown and McQueen [44]). In this case, we see that PH(cI = 0) as b1?)—* 1, but PH(cl <0)—* p1a1b~ic1in this same limit (Prieto and Renero [68]). Clearly, c1 cannot —*
equal zero for physically reasonable asymptotic behavior. Setting c~= 0 in (A.48) reduces it to the so-called shock wave “equation of state” (McQueen et al. [21]). For the r and/or other high-pressure phases of Fe, for example, c1 is apparently unresolvable by implication of how well the uv~ data are fit by a linear relation (Brown and McQueen [22]). Following Pastine and Piacesi [69], we substitute (A.46) into (A.40) to obtain 2~ a1 = (p[~P~]i~)’’ 115110 (A.49) .
Since the limits becomes
PH
~ p1, q —*0 and [~P~]—*0 are formally and physically equivalent to [~v] —*0, (B .35)
a1=0/0, an indeterminate form. However, since both [~P~]and ~ are analytic functions of ~ or PH via (A.48) and by definition, respectively, we may use L’Hopital’s rule to evaluate (A.49), i.e., 112 a.=~— , (A.50) (1 (d[~P~]/dpH)~ \p 1 (di~/dp~)/ PHPI or (A.51)
112~~P. a1 = (BHipH)
Using (A.46), (A.47), and their derivatives, we find, in a similar fashion, an expression for b~,i.e., b 1 = ~[1 + B~IpH~~p.],
(A.52) (A.53)
1p~(d2PHidp~) + 1. B~as(dBHidPH)= B~
Again in a similar fashion, we obtain 1 =
1 b 1(2 b1) + 2BH/dP~),
~—
—
as
~-~--—
(BHB~(,~P,
(A.54) (A.55)
(d 1p~(d3P
B 11B~= B~
11/dp~)(B~ 2)(B~ 1). —
—
—
(A.56)
Note that, up to this point, nothing specific has been assumed about the “initial” state ~ as { T1, P~,u~p1, s~,e1} to which (A.39), and so a1, b1, and c1, are referenced. If values for a~,b~and c1 are constrained by experiment, for example, then is the state of the uncompressed starting material, usually with T1 as 298 K, P1 = 0.1 MPa, and v1 = 0. But, as we have done in the text to write (4.19)—(4.22) and the relations that follow, we can use equivalent thermodynamic paths to “shock compress” the material from any initial or reference state. .~,
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B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
Bethe [67] has shown that, for shock compression of a non-heat-conducting ideal fluid, i.e. that represented by (A.2), (A.22), and (A.30), the initial (PH = p1) slope and curvature of the shock compression locus are equal to those of an isentrope referenced to the same initial state, i.e., BH!~P = K2,
(A.57)
~
(A.58)
as
K~,
where K2 and K~are the isentropic bulk modulus and its first pressure derivative referenced to (4.25) in the text, we may also relate B~to K’ (e.g. Pastine and Piacesi [69]), i.e., (BHB~)~PP = K2K~+ 2b1’y1
.~.
Via
(A.59)
.
In these relations, K2, K and K’ are the isentropic bulk modulus, and its first and second pressure derivatives at constant entropy. Note that K2, K~and K’~are equilibrium thermodynamic properties. In principle, (A.57)—(A.59) may be used as consistency relations between Hugoniot data and other types of compression data for appropriate materials. Substituting (A.57) and (A.58) into (A.51) and (A.52), respectively, we obtain ,2 (A.60) a1 = (K2/p~)’~ b
(A.61)
1=~(l+K~).
Doing the same with (A.59) in (A.55), we have c~=
~—
[2b~(2
—
b1
+
y1)
+
K,K~].
(A.62)
From (A.61) and (A.62), we see that, if c~as 0, then K,K’ as ~ (1 + K~)(K~ —7— 4~),
(A.63)
and this is the case for most materials from the experimental resolution point of view. We presume that values a’ and b~for a u—va relation have been constrained by experiment such that the shock wave “equation of state” (4.10) describes the (RH, PH) states of the high-pressure phase ~ along the Hugoniot. These states are also described by the “metastable” shock wave equation of state (4.23) based on ar and hr. Consequently, we may substitute (4.10) into (A.51) and (A.52) referenced to . r to obtain 3]112a~’, (A.64) ar as (1— 4.)[(1 + b~t~)/(1 b4) b~—1(1+ 4(1—4)b~+(2— ~)(b~)2~—1~ (A65
pr~
—
—
4 ‘\
respectively, where 4
(1 1
—
—
b~j)(1+ b~q5)
‘
p~’pr.Note that, from these relations, ar—> a 1 and br—+ b1 as p11—* p~,i.e.,
385
B. Svenda-en et a!., Optical radiation from shock-compressed materials and interfaces
as 4—*0. Note that we need p~to obtain ar and br, and so Kr, Kr’, and Kr” with cr=O, our initial goal. In the text, for example, we use relations analogous to (A.64) and (A.65) to estimate the isentropic properties of liquid Fe referenced to STP given in table 4.1. A.4. Impedance match
]
The balance of momentum at the boundary between adjacent target components, [~ = 0 and (A. 36), combined with the balance of momentum (A.22) across the shock front, form the basis for the impedance match technique (Rice et al. [20]), which is used to calculate the density, pressure, etc., of the high-pressure, shock-compressed states of each target component throughout this work. For completeness, we present here a brief development of the impedance match relations between any two adjacent target component layers A and B in the target. We orient these layers by having the shock wave propagate from A into B. The basis of the impedance match is found in the idea that the shock wave propagating from A into B accelerates, but does not shock, A material next to (i.e. within approximately a shock front’s thickness of the interface) the interface to “impact” with the B material adjacent to the interface. The compression of the interface then shocks A material next to the interface, and sends a shock wave propagating into B material. Imagining this compression takes place from zero pressure, the impact of A against B occurs at a velocity twice that of the material velocity of A, UHA in what follows). In the process, A material next to the interface (note that 13H U(j,) and v. as achieves the pressure, ~HB’ and normal material velocity, VHB, of B in order to maintain a material interface. This requires a net change of 2~HA 13HB in the material velocity of A interface material; this change is then the material velocity imparted to A material next to the interface by the shock wave. Again, we emphasize that A interface material is not shocked by the shock wave propagating through A; rather, it is shocked as it “impacts” B. Hence, the material in A adjacent to B at the interface between these layers experiences a different shock compression history than the material in the interior. Since we do not measure the shock wave velocity in the radiation experiments, the basic aim of the impedance match for our purposes is to calculate the normal component of the shock-induced material velocity in B as a function of (1) that in A, and (2) certain material properties of each layer, as the shock wave propagates from A to B. As emphasized above, in this impedance match, we are dealing with the A layer material “right” at the interface, which is not shocked by the shock wave propagating through the A layer, and its analog in B. To begin, we have from (A.22) the pressure generated in A interface material and B material as a result of shock compression, i.e., —
~HA = ‘~iA+
PIA(uA
~HH = 1~iB+
PIB(UB
—
UIA)(vHA
— ~IB)(~HB
(A.66)
— ~
—
(A.67)
~1B)
(with P 1 as p and ~H as P ). Again, we emphasize that (A. 66) represents the shock-compressed state of A interface material after it has impacted B interface material; this state is in general different from the shock-compressed state of A material in the back of the interface. To proceed further, we adopt the constitutive assumption (A.39) for both layers, 13iA = alA + blA(vHA CIA) + clA(UHA ~1A) , (A.68) 2 +
—
—
—
u~ 3= ~
+
bIBVHB
+
cIBUHB.
(A.69)
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B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
In (A.69), we have used the fact that ~ 0 in the experiments. From the discussion above we have = 0 when we require VHA = 13HB for a material interface. Putting that 13HA 13iA = 2VHA VHB, or this along with (A.68) into (A.67), and assuming =0 = P 1~,we obtain —
—
13iA
~jA
2vHA
—
~HB) + PIAbIA(2VHA
—
VHB)2
(A.70)
+ plAcIA(2vHA — vHB).
plAaIA(
~HA
Likewise, substituting (A.69) into (A.67), we have = PIBaIB13HB + PIBbIBVHB + pIBcIBvHB
~HB
And since
=
~HA
~HB
plAaIA(213HA
—
(A.7l)
.
for a material interface, these last two expressions combine to give ~HB) + PIAbIA(2VHA
—
vHB) + PIACIA(2VHA
= pIBaIBvHB + PIBbIBVHB + pIBcIBvHB
Rearranging this into a polynomial in
VHB,
—
~HB)
(A.72)
.
we obtain (A.73)
UHB+aUHB+bvHB+c=O,
a
as (PIAbIA
b as 12p1Ac1Av~A
PIBbIB) — 6pIAcIAvHA PIAcIA — PIBcIB —
2p~~(a~ + 2bIA13HA PIAcIA
—
—
(PIAaIA + 4PIAbIAUHA + plBalB) PIAcIA — plBclB
+ 4cIAv~A)vHA
plBclB
We may solve (A.73) for 13HB by first substituting 13HB = w ~a into (A.73) to get w3 3pw 2q = 0, with p as ~(a2 3b) and q ~(9ab 27c 2a3). Next, we set w = \/75 cos 0 and use the trigonometric identity 4 cos30 3 cos 0 = cos 30 to obtain cos 30 as q/p~./~,and so 0 as ~ cos’(q/pV~). Since 30, 30 + 2ir, and 30 + 4ir all have the same cosine value, we have w = 2~/~x {cos 0, cos(0 + 21T/3), cos(0 + 41T/3)}, i.e. three roots for w. Consequently, we obtain —
—
—
—
—
—
—
13H5
=2-s/J5X
{cos0,cos(0 +21T/3),cos(0 +4ir/3)}
—
(A.74)
~a,
representing three roots, i.e. possible values, for VHB. Little of a general nature may be said regarding this solution. The only general guideline we have to choose the appropriate root is the physical requirement that 0HB be nonnegative real, just as the coefficients a, b, and c are real. Beyond that, we must use physical plausibility to make this choice. For the particular purpose of the experiments, we are usually justified in assuming PIAcIA as PIBCIB. In this case, (A.72) reduces to
(A.75)
VHB+avHB+b=O, =
a
—
12pIAcIAv~A
—
(PIAbIA
(PIAaIA + 4PIAbIAUHA + PIBaIB) — PIBbIB) — 6PIACIAUHA
b
+ 2bIAVHA + 4cIAv~A)vHA (PIAbIA — PIBbIB) — 6pIAcIAvHA
= 2P~A —
B.
Svendsen et al., Optical radiation from shock-compressed materials and interfaces
387
and solutions (A.76)
VHB±—2(a±Va4b).
Since a and b are, in general, real, we require a2
4b in order that
addition, we require VHB_*O as UHA~_*O,which gives us 131-IB
13HB
be nonnegative real. In
as 13HB~i.e.,
—~(a—Va2—4b),
(A.77)
since b—*0 as Using (A.77), we now write the impedance match for the particular type of targets used in the experimental studies presented in this work. These consist of (1) a metal driver plate and (2) one or more “sample” layers (e.g. the film/foil and window layers in fig. 2.1). The driver plate is usually impacted by a metal flyer plate traveling at a velocity Vim~Letting the flyer plate be layer A, and the driver plate be layer B, we have, from (A.77), (A.78) where now a=
—
plFalF
+ 2PiFbiFVjm + PIDaID
PIFbIF
with
2~HF
=
Vim
—
b
= PIF(aIF + bjFVjm) Vim
PiD’~iD
PIFb~F
—
PIDbID
and c
1~= 0. In these expressions, the subscripts “F”, “D”, and “S” stand for the flyer plate, driver plate and sample layers, respectively. For any two layers in the target, such as between the driver plate and sample layer, we have (with c1~~ = 0)
(A.79)
2—4b), VHS=—~(a—Va a=
—
p~DaID+ 4PIDbIDVHD + p~ PIDbID p~~b~ 5a1~
2P~~’~~D + 2bIDVHD)UHD
1-’ts 5 PIDbID P~s for the normal material velocity in the sample layer. The impedance match between any remaining layers has this form, in the linear approximation. So, given the impact velocity, 131m’ of any experiment, and the appropriate material parameters of each layer, we may calculate the material velocity achieved in each layer due to shock compression as constrained by the requirement of material interfaces between layers. From these velocities follow the shock wave velocity via the constitutive assumption of a shock velocity, material velocity relation [e.g. (A.39)], and pressure via (A.22), of the first shock state achieved in each target layer. —
b
=
—
Appendix B. Equilibrium thermodynamics and shock compression The calculations discussed in the text for Fe—Fe—Al 203 and Fe—Fe—LiF targets are based on equilibrium thermodynamic models for the Helmholtz free energy, F( T, p), of the solid and liquid states of Fe. As discussed in the text, this model is based on (1) a Debye model for the harmonic
388
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
contribution, (2) the low-temperature (T ~ TF, the Fermi temperature) electronic contribution, and (3) high-temperature (T> TD, the Debye temperature) anharmonic contribution, which we combine for simplicity, since they are of the same order in T. The liquid state model is presented elsewhere (Svendsen et al. [48]). In the last two parts of this appendix, we detail certain relationships between isentropic and Hugoniot states using this model, and a method to recenter experimental U—V relations to the STP density of the high-pressure phase, both of which are used in the calculations. B.1. Solid-state equilibrium thermodynamic model For a cubic or isotropic solid material subject to an isotropic state of stress, the combination of the Debye model for the harmonic contribution along with the high-temperature anharmonic and lowtemperature electronic contributions provides an expression for the Helmholtz free energy, F( T, p) (e.g. Wallace [46] sections 5, 19, 24; eq. (4.33) in the text), i.e., F(T, p)
as[~1(p)
+
~yRT~(p)] + 3vR[ln(1
—
e~t))
—
~ED(~D)]T
+
A2(p)T2
—
~F(p)T2.
(B.1)
Note that F has units of J / mol. For simplicity, we neglect possible bandstructure and electron—phonon interaction contributions to F in writing (B.1). Relation (B.1) is correct to 0( T 3) in the anharmonic contribution to F, and to all orders for the harmonic contribution in the context of the Debye approximation. This slight inconsistency is due to the unavailability of a tractable anharmonic model, analogous to the Debye model, for the materials of interest. Also, for Fe, we are guided by the results of Andrews [42], who was able to fit various data on the thermostatic properties of the a- and c-phases with a Debye model for the harmonic contribution to F, ignoring anharmonicity altogether (although anharmonicity may be reflected in the value of his F). As discussed in the text, the anharmonic coefficient A 2 ( p) is dependent on the particular model chosen for the pressure and temperature dependence of the anharmonic phonon frequency spectrum. Since we do not have such a model for the materials of interest here, we simply combine it with the electronic contribution to form Ii(p), as given in the text. On this basis, (B.1) provides the means to a rational parameterization of the approximate density and temperature dependence of a number of solid-state properties, such as the pressure, i.e., 2, P(T, p)as p(9F/ô ln P)T P(0, p) + 3vRpyDED(~D)T+ ~pwQT P(0,p)asp(dtP/dlnp)+ ~VRPTDYD, where
YD
(B.2) (B.3)
is the lattice Grüneisen parameter in the Debye approximation, as defined in the text. From
(B.l), the molar entropy is given by S(T, p)as —(c9F/ôT)~ ‘3vR[~ED(~D)—ln(l —e~)] + LiT.
(B.4)
The isothermal bulk modulus is given by KT(T,
-) as
~
e~~lT+ ~pw(l—w)QT2, (B.5)
B. K(0,
P(0, p)
p)as
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
+
389
p[p(d~I3idp) + p2(d2a~Pidp2)]+ ~ziRpyDTD(yDqD), —
q~as_dlny~/dlnp,
(B.6)
which is assumed constant. The molar heat capacity at constant volume (density) is CV(T,p)as_T(ô2F/ôT2)V_3VR(4ED(~D)_ ~
(B.7)
1)+LiT.
The change in pressure with temperature at constant density is given by, from eq. (B .3), pycv
(~Pi~T)~ = aKT
=
=
3v(R/M)PYD(4ED( ~D)
—
e~-~ 1)
+
pw([liM)T,
(B.8)
with c~= C~/M.From (B.7) and (B.8), we have ycv = yDcv +
(w
—
(B.9)
YD)(L/M)T,
and so Y
7r
+
(w
—
(B.10)
YD)(Li~V)T,
which is very weakly temperature dependent above TD since w as y~and 12 ~ 3 yR. Other properties given by a ratio of the derivatives of F(T, p) include the coefficient of thermal expansion, a(T, p)
=
pyc~iK~,
(B.1l)
the isentropic bulk modulus K2(T, P) = KT
+
(B.12)
pyc~yT,
and the specific heat capacity at constant pressure, (B.l3)
c~(T,p)=c~(1+ayT). These are the solid-state model properties used in the text. B.2. Connection of isentrope and Hugoniot
To use the model as detailed above, we need to calculate the change of certain model properties, such as P, with density at zero temperature. So, we relate the zero-temperature model properties to the known change in density along an appropriate isentrope or the Hugoniot, as follows. From (B .3), we have ~
with
~Ds as
2PWQTSH,
TD/TSH. P(TSH, p) is
given
by eq. (4.32) in the text, i.e.,
(B.14)
390
B. Svendsen a a!., Optical radiation from shock-compressed materials and interfaces
P(TSH, PH) =
Pj
+
PHYH AeSH + (PH/P~)[l
—
(l +
27H)?)ls](1’H
—
P1),
(B.15)
and 1~eSHis calculated numerically, as discussed in the text. In a similar fashion, we have, from (B .5) and (B.12), 3’YD)ED(~D)T
KT(O,
p) = K2(T, p) p-yc~T 3v(R/M)pyD(l qD —9v(R/M)pyD e~~lT— ~p(l—w)w(12/M)T2. —
—
—
—
(B.16)
To get an expression for K 2(T, p) along the Hugoniot, we follow McQueen et al. [21] in equating an infinitesimal change in specific internal energy (SIE) along the Hugoniot with one along an equilibrium thermodynamic path, as follows. From the first law of equilibrium thermodynamics, we may write, at a given density Pr’ the relation e(s, Pr) = e(sr, Pr) +
f
T(s*, Pr) ds*,
(B.17)
where e(s, p) is the SIE and s is the specific entropy. Since we have assumed the shock-compressed state is one of thermodynamic equilibrium, we may set Pr = PH’ Sr = SH, and write de(s, PH)
=
de(sH, PH) + d(J T(s*, PH) ds*),
(B.18)
giving us an expression relating an infinitesimal change in SIE along the Hugoniot, de(sH, PH)’ to one at the same density but at another specific entropy. From eq. (4.21) in the text, we have another expression for de(sH, PH), i.e., (B.19) PH
Substituting this into (B .18), we have ~
T(S*,pH)ds*).
(B.20)
Now assuming s = s(P, p) [i.e. e = e(P, p)], we have 1 K Tds=—dP_--~-dp. P7 P7
(B.21)
Putting this into (B.20) and rearranging, we obtain K~=
7H~— ~7H(~H
+
Pr) +
p1
[1—(1+ ~yH)?)p]BH YHP~~ PH —
(J
T(s*, PH) ds*),
(B.22)
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
391
BHaspfld[~PI]/dpH=(1 —n~)dPH/d?)~, where YH
as
7(~H,PH). Letting P
KS(TH, PH)=(PH/Pr)[1
—*
~H
and ~
--*
(B.23)
~I1’
we obtain the desired expression,
—(1+ 27H)?)It]BH+ 27H(”H
P1).
(B.24)
In the text, we further assume YH as Thu’ i.e., that y is a function of density alone.
Appendix C. Isentropic release and reshock Considering the shock-compressed material as a thermodynamic “system” and the “lab” as its “surroundings”, a balance of energy implies that any infinitesimal change in the specific internal energy of the system, de( q, w), is due to the difference between the net amount of heat transported into the system from the surroundings, q, and the net amount of work done by the system on its surroundings, w. If we assume the material to be internally in thermodynamic equilibrium, then de = T ds P dU and any infinitesimal change in specific entropy of the system is given by —
(C.1)
ds=~q—~(w—PdV).
Adopting the idea that heat transport in or out of the target is insignificant on the time scale of release, we assume that the release path is adiabatic (q = 0). Further, we assume that the mechanical work done by the system during release is entirely reversible (w = P dv). In this case, the release path is both isentropic and adiabatic. If a phase transition occurs during release, the constraint of isentropic release in turn places constraints on the phase transition. Consider a first-order transition from the shock-compressed phase 13 to a release phase ~r.In this case, the total specific internal energy, e, total specific entropy, s, and total specific volume, v,. of the two-phase system may be written in the form (C.2) where x is the mass fraction of ir, and 4i as {e, s, v}. As with a single-component system, an infinitesimal change in specific internal energy of the two-phase system, de, is balanced by the net heat flow into the system from the surroundings, q, minus the net work done by the system on its surroundings, w. Assuming that each phase is internally in thermodynamic equilibrium (i.e., temperature, pressure and composition are homogeneous within the phase), an infinitesimal change in total specific entropy of the two-phase, single-component system is, from (C. 1) and (C.2), T~ds = q + ~(T~
—
T’~)ds” + [(gP g~)+ —
s1T(T~—
T”)] d~
—
Wjr~
(C.3)
where P~d[(1
T]
x)u~] P d[XV’ is the nonrecoverable work done by the system on the surroundings, and g is the specific Gibbs free Wir as
—
—
—
B.
392
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
energy. The second term on the right-hand side of (C.3) is the entropy produced by heat flow between phases, while the third term is that produced by mass exchange between phases. A sufficient, but not necessary, set of conditions for isentropic (ds = 0) release through the mixed-phase region is then, on the basis of (C.3), (1) adiabatic release (q as 0), (2) mechanical equilibrium (P~= P”) in the system plus reversible (P dV) work on the surroundings (P = as P’~),(3) internal thermal (T~= T”), and (4) chemical (g6 = g’~)equilibrium. Clearly, from (C.3) we see that even if release is isentropic through the mixed-phase region, it is no guarantee that the phase transition will occur in thermodynamic (i.e. thermal, mechanical and chemical) equilibrium. We choose to satisfy the constraint of isentropic release by assuming the conditions discussed above consistent with this, i.e. (1) release is adiabatic, (2) all work is recoverable, and (3) the phase transition occurs in thermodynamic equilibrium. On this basis, we need an expression for the change in pressure and temperature across an isentropic phase transition. With s = s(P, T) for each phase, an infinitesimal change of s in the mixed-phase region is, from (C.2) with 4 = ds = (c~+ x Ac~)dlnT— (a~V6+ x Aav)dP
+
As d~,
(C.4)
where c~is the specific heat at constant pressure, V 15 the specific volume, a is the thermal expansion, and A4 as ~ is the jump of any quantity 4 across the mixed-phase region. For an isentrope through the mixed-phase region, ds = 0. Putting this condition into (C. 4), and noting that pressure and temperature do not vary independently along the equilibrium phase boundary, i.e., At dP as As d TPB, we have —
[(c~ + ,y Ac~)AV—(a6V~+ x Aav)TPB As]dP
+
As TPB ~ d,~k’= 0.
(C.5)
We now propose to solve (C.5) for x and apply it to the problem of isentropic release when the release path traverses all or part of a mixed phase region, as discussed in section 4.2 in the text. Let 11(P, x)
as
(c~,+ x Ace) Av
X(P)
As TPB As.
—
Tp~(a~V~ + x Aav) As,
(C.6) (C.7)
Then (C.5) may be written H(P,~)dP+X(P)d~=0.
From (C.6) and (C.7) we have
HX(P, x) as (aHh)x)~= Ac,, Av X,,(P)
as
—
TPB
Aav As,
(oXioP)~= As (dTPB/dP) As
+
2 As TPB(dAs/dP)PB
(C.8) =
[Au + 2TPB(dAs/dP)PB] As. (C.9)
Note that both of these partial derivatives are functions only of pressure along the phase boundary.
)
B.
Svendsen et a!., Optical radiation from shock-compressed materiaLs and interfaces
393
Since at any point along the phase boundary P as P~= P~,we have (dAs\ ~—J \ dP /PB
Ac~Au ——--Aav. TPB As
(C.10)
Putting this into (C.9), we have
X,,(P) = (AU
—
2TPB Aau) As
+
2 Ac~AU.
(C.11)
Now, (C.5) will be an exact differential if X,., as lix. Since, from (C.8) and (C.1l), this is clearly not true, (C.5) is not an exact differential equation, in its current form. However, we may attempt to put it into such a form by solving pii(P, x) dP + p~X(P)d~ Ø,
(C.l2)
where p~is the integrating factor. Putting these into the exactness criterion, we require (~H)~ as (~X)~.
(C.13)
Expanding (C.12), we obtain
H/~~XILp+(H~Xp)I20.
(C.14)
Note that (H~ X~)is a function of P only. In this case, we set p~= 0 and solve —
(H-X ‘~
X
~,
(C.15)
to obtain the integrating factor,
~(P) With
as
p~P),
exp{f
~-
[AaV
-
TPB
+
j]
dP}.
(C.16)
(C.5) in the form (C.12) is now exact. On this basis, we may solve (C.5) as follows. Define
V’~(P,x) as ~(P)H(P, x),
(C.17)
~
(C.18)
x)
as
p~(P)X(P).
Integrating (C.17) with respect to pressure at constant composition, we obtain
~(P, X)J~(P*)H(P*,X)dP* + g(x), where g(,~’)is at most an arbitrary function of composition. Putting this into (C. 18), we have
(C.l~)
B.
394
~(P)
~(P)X(P)
Svend.sen et a!., Optical radiation from shock-compressed materials and interfaces
=
J
~(P*)~(P*,
x) dP* + g~(x).
(C.20)
Solving this for g~ ( x), we find
I
g~(x)= ~(P)X(P) ~(P*)H(P* x) dP*. Even though it appears that g(,y) is a function of —
pressure,
(C.21) it cannot be, by its definition (C.20).
Further, since HX(P,
x)
=
Ac~AU
(C.22)
TPB Aav As
—
is actually not a function of x~we have
g(x)—xg~(x),
(C.23)
and so
~(P, x)
=
~~(P)X(P)
+
J
~(P~)[H(P~ x)
—
Xhjx(P*)] dP*,
(C.24)
with H—H~=c~Au— TPBa~u~As,
(C.25)
from (C.7) and (C.22). Putting (C.25) into (C.24), we obtain 1I~(P,x) =
1(P)
as
J
~p~(P)X(P)
~(P*)[T
Note that ‘I’(P, x)
is
—
1(P),
a~u~ As
—
(C.26)
c~AV] dP*.
equal to some constant since
~P~dX+~1tpdP0
(C.27)
by definition (C.12) above. Consequently, (C.26) gives us ~1’(P,x) = ~~(P)X(P)
—
1(P)
=
constant.
(C.28)
As an initial condition, we assume that the mass fraction of ‘rr in the Hugoniot state is XH as x(PH). In this case, (C.26) gives us
~P(P,x) as W(PH, XH)
B.
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
395
or
~(P)
=
~1
~(P)X(P) (~(PH)x(PH)xH
~(P*)(T
aV
As
—
c~AU) dP*)
(C.29)
This last relation is used in section 4.2 to estimate the effect of possible melting transitions on the release process.
Appendix D. Energy transport in the target In this appendix, we (1) derive conditions under which radiation and/or conduction may be an important means of energy transport in the shock-compressed or released states of the target components, and (2) establish energy balance relations for the target components to be used in appendix E and the text. We attempt (1) via a dimensional analysis of the local energy balance in a radiating, conducting target represented as a rectangular Cartesian continuum with material properties assumed to be isotropic and homogeneous in the reference (deformed) state. In constructing this energy balance, we assume that the radiant energy density and radiation stresses are negligible in comparison with the corresponding thermomechanical quantities. We also assume an equilibrium caloric constitutive relation for the specific internal energy of the relevant deformed state of each target component. With the deformed state as the reference state, then, we may assume, for example, that the heat flux q is given by the classic Fourier relation in an isotropic medium qas—kVT,
(D.1)
where k is the thermal conductivity. Under these circumstances, the local balance of energy in the neighborhood of the (deformed) reference state r is represented by Prcpr
da-T(x,
t) =
kr V2T(x, t)
—
div h(x, t)
+ p(x, t).
(D.2)
In this relation, T is the absolute temperature, h is the radiant energy flux, and we have combined terms for the specific internal energy supply and mechanical dissipation rates to form ~(x, t). Also, Pr’ cp,. and kr are the density, specific heat at constant pressure and thermal conductivity of the reference state. Note that, in (D.2), time is judged relative to the beginning of the reference state, and s9a- as 81ôt. Of the terms in (D.2), neither div h, the divergence of the radiative flux, nor ~, the specific internal energy production rate, is specified. To find an expression for div h, we first need to discuss radiative transport in the target. All radiative transport models discussed in this paper are founded in geometric optics and the classical equation of transport [eq. (D.3) below], both of which assume an optically isotropic propagation medium (i.e. one with a uniform index of refraction). The form of this equation we use here presumes that (1) all processes affecting the observed radiation intensity (scattering, absorption, etc.) are independent of the intensity of the sources and (2) any scattering is elastic and isotropic. Under these conditions, the change in the quasistatic spectral intensity, ii(s), at a point p(x) in the direction s is given by (e.g. Siegel and Howell [15], eq. (14-4))
B.
396
Svendsen et a!.. Optical radiation from shock-compressed materials and interfaces
f
~
i~(w~)dw~.
(D.3)
This is the radiant energy transport at a point p(x) per unit time, per unit projected surface area normal to s, per unit wavelength interval dA about a single wavelength A, per unit solid angle in a single direction s. Note that the subscript A denotes a spectral quantity. i is quasistatic because we assume it is not explicitly dependent on time, but only implicitly so through temperature, etc. In (D.3), V as ~ 0~A= o~ (A, T, P) are the spectral absorption and scattering coefficients, respeca 5 as a5and (A, T, and tively, ~e P) = ~a(A, T) is the spectral emission intensity, all along s. The first term on the right-hand side of (D.3) is the loss of intensity by absorption (including the negative contribution from induced emission), the second is the gain by all emission processes except induced emission, the third is the loss of intensity by scattering, and the last is the intensity gained by scattering into the s direction from the solid angle w. Note that we assume, in writing (D.3), that the wavelength of the radiation is not changed as a result of these processes. If we integrate (D.3) over all solid angles and wavelengths, assuming any scattering to be isotropic, we obtain div h at a point p(x) (e.g. Siegel and Howell [15]), i.e., div h
=
4~Ja5(A )[i(A)
—
~(A*)] dA*.
(D.4)
In this relation, the first term represents the rate of emission of radiation per unit volume in all directions, while the second, given by
f
~(A)as
i~(A,w*)dw*,
a =4,r
is the radiation intensity scattered from all directions into p at wavelength A. In principle, we could substitute (D.4) into (D.2) and try to solve the resulting nonlinear integrodifferential equation for the temperature field in the medium of interest. However, here we want only to establish the magnitude of div h relative to other terms in the energy balance, eq. (D.2). To judge the relative magnitude of the terms constituting (D.2), we render them nondimensional (ND) by the following transformation: {x,
t,
T, h,
~}—* {Xr~, trT~ Tr +
ATrl~hrh~~r~}
(D.5)
In this transformation, 6’ and Tr are the ND and reference (e.g. Hugoniot or release-state) temperatures of the material, while A T~is the difference between Tr and some “maximum” possible temperature such that 19 ~ 0(1). For example, if Tr is the release-state temperature, A Tr represents the difference between it and the reshocked-state temperature at the same pressure (see text). Substituting (D.5) into (D.2), we obtain 219 + —
~
V
~—
div h
= Dar~,.
(D.6)
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
397
In this expression, Per as X~itrKr is the Péclet number, Kr as kr/Prcpr is the thermal diffusivity, BOr as Pr~~Pr A Trihrtr is the Boltzmann number, and Da,. as ~ A Tr is the Damköhler number, of state r. Per, BOr and Dar are, respectively, the ratios of (1) free enthalpy flux to conductive flux, (2) free enthalpy flux to radiative flux, and (3) generalized internal energy supply rate to free enthalpy flux. We are particularly interested in Per and BOr~since their respective magnitudes will control the relative contributions of conduction and radiation to the energy transfer in the interior (away from the boundaries) of any layer of the target model. To obtain the magnitude of BOr~we first require an expression for hr, which may be obtained from (D.4). First, we assume ~e(A) is given by = jAb(nA,
A, T)
(D.7)
,
where tAb( A, T) is the Planck function and n5 is the index of refraction of the medium around p. Then, defining the Planck mean absorption coefficient (e.g. Siegel and Howell [15]), —
.f~° a5(A,
T, P)isb(nA, A, T)dA
(D.8)
1Ab~!~k’ It, ~ T\,1~ I )U/t
JO
and the incident mean absorption coefficient, f~a
(a)~~(T, P)
1A~dA 5(A,T,P)i~(A)dA as
coo
-
as
J
—
L 5(A) dA,
(D.9)
£5~~
and W/m2 noting K4that, if Stefan—Boltzmann n5 is independent constant, of A, j~° isb(nA, dA = in n OSBT , where 108 is the we may A, putT)(D.4) the form div h
41T[(a)~
as
0SB as 5.6696
x
2ff~~T4(a) —
1n
(D.10)
1~fl.
Nondimensionalizing (D. 10) with the appropriate transformations in (D.5) plus i—* i i~we have div h =
[(a)~~(1 +
where ~,as nro~suTr, hr as ~ ber, we obtain Bor
4lTartrnrtrsBTr pc AT =
r Pr2
r
~
~r6’)~
—
(a)111fl, and
(D.11) ~r
=
ATriTr~Substituting hr into the Boltzmann num-
(D.12)
From (D.6) and (D. 12), we see that radiative transfer will be an important means of energy transfer in an optically thick [Bo,.(a,.—*cc)--*0]and/or high-temperature medium, but not in an optically thin [Bor(ar—* 0)—* cc] medium, all other parameters being finite. At a boundary between two target components, we have a slightly different energy balance to consider. If we assume that the boundary is material and does not contribute to the balance of energy across it, the local balance of energy across the boundary between layers A and B takes the form
398
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
(D.13) where v is the “outward” normal unit vector to the interface. Using (D. 1) and the definition of hr given above, this takes on the ND form (VøA_~V19B).Vas~~~_ hA~v—~—
hBv,
(D.14)
B where ~ as kB ATB
S0
=
xA/kA
ATA XB, and Sk0 is the Stark number of layer 0, given by
k0AT0Ix0 2
(.)
4.
4ira0x0n0o~5T0
This parameter is a measure of conductive to radiative flux across a layer x0 thick with effective temperature gradient A T0/x0. Estimates of the parameters in these relations for the high-pressure states of the metallic DP, FL and dielectric SW are given in table D. 1. Of the parameters entering into Per, BOr~ and Skr~we note that Pr’ cpr, Tr~ATr, texp and ~1r are relatively well known (i.e. to within a factor of two) through the impedance match and equilibrium thermodynamic shock compression/release calculations discussed in the text. Most uncertain of all material parameters constituting Bo,. and Per, and so Skr as Bor/Per, are the thermal conductivity, k~, and the absorption coefficient, ar, of the high-pressure state of each layer. For perfect crystalline nonmetals k ~ lIT, while for metals k is constant at high temperature (e.g. Berman [59]; for minerals, see Roufosse and Klemens [60]), implying that the values of kr in table D. 1 are upper bounds if these dependences are relatively insensitive to pressure. If we assume that k for metals is dominated by its electronic component, ke~we can use the Wiedemann—Franz—Lorenz (WFL) relation [eq. (5.7) in the text] to estimate k from electrical resistance data, or some assumption on Pe’ since the relevant data are relatively scarce. Bridgman [70] investigated the change in electrical resistance of many statically compressed materials, including iron, but only at low pressure (s10 GPa). Keeler [39] investigated the change in the electrical resistance of shock-compressed copper and iron up to 140 GPa. He found that
Table D.1 Order of magnitude of parameters parameter
k,a)
T~ n~ a’~
metal
4
dielectric
iO~
i03 i0 102
101
i04 iol 1
iO~ iO~ 1
>106
10
SI units
kg/rn3
J/kgK W/mK
K K m~ rn
1o~ Pc, Bo,
<1 10
Sk,
10 ~ STP values.
lo~ <106
io~
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
399
Pe of shock-compressed Cu decreased from as16.7 to 5.6 nfl m with pressure up to —100 GPa; a datum
at 140 GPa implies that the resistivity of Cu reaches a minimum between 100 and 140 GPa and then increases to —0.83 nfl m at 140 GPa. As for Fe, the data imply that its electrical resistivity decreases from —2.5 to 0.47 nfl m between 13 and 140 GPa (above the a—* s transition). In light of (5.7), these trends imply that the thermal conductivities of shock-compressed Cu and Fe increase with shock pressure, against the high-temperature expectation expressed above. Consequently, we may assume with some basis that the STP value of k provides a lower bound to the high-P, T value of the metallic target components, k,.. The thermal conductivity of dielectric target components is dominated by the lattice contribution, k~, at high pressure; and we may estimate k from the lattice thermal-conductivity model of Roufosse and Klemens [60], who argue that the acoustic branches of the phonon spectrum dominate k~.Using this assumption, they arrive at an expression for k~,i.e., k~= (B’IT)[~(T1IT)112 + ~(T/T1)]
(D.16)
,
with T
‘l
=
213 _________ 1/6 25/3 2
n
(31T
)
Ma U 2 kBYP
‘
1
D’ ‘-‘
Ma
_________ —
2
1/6
3
2/3
ir
7/3
2
a n
1/3
2
where n is the number of atoms in the unit cell, Ma is the atomic mass, U, 1, is the velocity of sound, kB is 3 is the atomic volume, and is the acoustic-phonon Grüneisen parameter. Boltzmann’s constant, a This relation is consistent with the k ~ 1 / T expectation at low pressure. However, if the effect of pressure is to increase k, then k may change little from its STP value. Arguments in favor of this are given by, e.g., Roufosse and Jeanloz [71], who find that various two-body interatomic force models appropriate for halides predict an increase in k~with density and a decrease across polymorphic phase transitions with an increase in coordination. We use (D.16), (D.17), the solid-state equilibrium thermodynamic model presented in appendix B, and the equilibrium shock compression relations presented in the text to estimate high-P, T values of ke for metals and k~for dielectrics in our targets. These calculations, discussed in the text, are consistent with assuming the STP value of k for metals and nonmetals is a lower bound to the high-P, T values of k for these materials. As for the absorption coefficient, we expect the STP values given in table D. 1, like those for k, to be lower bounds, since absorption in the optical band is most likely to increase with pressure and temperature (e.g. Siegel and Howell [15]). Only a few initially transparent materials, such as A1 203 (Bass et al. [6]), retain their original transparency upon shock compression to high pressure (~250GPa). From the parameter values given in table D.1, we expect B0F ~ i0~m/aF and Bo~ 1010 m~/a~ for the FL and SW, respectively. Since we have no upper bounds on a~7and a~.,we cannot really say that radiation 7will important within the FLtransport and SW, should but it seems unlikely. What we can to saythe is m’never and bea~ ~ 1010 m’, radiative not contribute significantly that if a,. ~ i0 energy balance within the FL and SW, respectively, on the time scale of the experiments. Note that this bound for aF is probably underestimated, since we have assumed a rather large value for Tr in the DP or FL (table D.l). As for conduction, we have, from table D.1, PeF ~ 1012 m2 x~and Pe~~ 1012 m2 x~,and this implies that conduction will be significant over a length scale of s106 m in the DP, FL and/or SW. On
400
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
this basis, it is likely that conduction will contribute to the energy balance in both the FL and SW on the time scale of the experiment over a length scale equal to the thickness of the FL, since we expect k~, and especially kF, to increase with pressure and temperature (i.e. during shock compression). From all this, we believe that the values of BOr and Per given in table D. 1 are upper bounds to all relevant values of these parameters at higher pressures and temperatures. In order to emphasize the uncertainty of k~,a,. and the governing length scales, XF and x~,,of the FL and SW, respectively, we write (D.6) for the FL (or DP) and SW, and (D.13) for the FL—SW interface, using the values for the better constrained parameters from table D. 1, in the following forms (with A as FL and B as SW): —
tlr6’w
—
10’4(kF/x~)V219F+ 107a,. div hF
=
DaF~,.,
(D.18)
10~3(k~/4)V26’~ + 10~°a~ div h~= ~ WXF
k
.7
aFxF
3
V(9~~v+10j~—hFv10
FXW
F
(D.19)
xFxWaW
k
h~’v.
(D.20)
F
For a given layer to be, in effect, spatially infinite with respect to a given process (i.e., the boundaries have no effect on the process), the length scale should be, roughly speaking, about an order of magnitude less than the layer thickness. The conductive length scale is x,~ 0,,as ‘~/‘(rtexp,while that for radiation is Xrad as a’. The STP values of these length scales, given in table D.l, when compared to x,. of each layer, imply that both the DP and SW are infinite with respect to conduction, but only the DP is with respect to radiation. This implies that any conduction in the DP and SW, and radiative transfer in the DP, will be quite localized on the 1timel0~ scale thethis experiments. will shock pass essentially m of In case, only Radiation the interface, front and unhindered through the SW for ar ~ x USW free surface will significantly affect the radiation intensity. We consider the effect on radiation of propagation across the interface, SW, shock front, USW and its free surface in appendix F. By the same reasoning, we conclude that any conduction effects on energy transport affecting the observed radiation field will be confined to a region near the FL—SW interface on the order of the FL thickness. On this length scale, heterogeneous heating at the DP—FL interface and/or phase changes between the DP—FL and FL—SW interfaces may also contribute to the temperature field in the neighborhood of the FL—SW interface. In light of all this, we assume that XF and x~,are given by the corresponding values of ~ for the FL and SW, respectively. Putting these into (D. 18)—(D.20) and reducing them, we obtain ~.
—
—
102kF V219,. + 107aF div hF
=
DaF~,
(D.22)
10~k~ V219~= ~
V19F~v=(kw/ kF)V19F~v+ 105(a,./k,.)h,..
(D.21)
ii,
(D.23)
where we have eliminated the radiative terms for the SW in (D.22) and (D.23), since a~would have to be on the order of ~ m~before radiative transfer is important in the SW and across the FL—SW interface, regardless of the value of x~,.If a~is much larger (e.g. 108 m’) and kF much smaller (e.g. 10) than their values given in table D.1, radiative transfer could influence energy transport in the DP, FL, and across the FL—SW interface. However, for lack of better information, we adopt the values in
B. Svendsen
et a!., Optical radiation from shock-compressed materials and interfaces
401
table D. 1, thereby assuming that the thermal conductivity of the DP, FL and SW, and the absorption coefficient of the DP and FL, are effectively unaffected by compression/release. In this case, (D.21)—(D.23) become Per’ V2ØF
—
DaF~,
(D.24)
Pe~,’V2ø~,= ~
(D.25)
V19D.Vas~tV19W.V,
(D.26)
—
where {PeF, Pew) 1 (with XF x~) and js 10_i to 1. In this case, radiative transfer does not contribute substantially to energy transport in any part of the DP—TW system. So, we are left to investigate the effects of conduction at the interface through relations (D.24)—(D.26) in appendix E, and the effects of radiative transport through the FL—SW interface, SW, shock front, USW and its free surface by way of a model based on (D.3) in appendix F. ‘-—
Appendix E. Conductive transport model The scaling arguments presented in appendix D provide some idea of conditions under which conduction and/or radiation may be an important means of energy transport in the interior of each layer of the target, and across the boundaries between the layers. With some basis for believing that we are in the range of conditions where conduction dominates radiation as a means of energy transport, we decouple these processes and treat them separately. In this appendix, we establish a simple conduction model for the target geometry. We adopt the framework of Grover and Urtiew [9], who assumed that (1) significant conduction takes place only along the direction of shock propagation (i.e. is one dimensional), and (2) shock compression transfers energy to the material much faster than it can be conducted away. This last assumption allows us to treat the DP and SW as thermal half-spaces. Choosing the initial thickness of the FL, d, as the governing length scale, we have the ND energy balance for the compressed/released/reshocked DP (—cc < ~ < —1, assumed the same as FL), compressed/released/reshocked FL (—1
~6’A(~, r)= —PeADaA~’A(~, r) (r>0),
(E.1)
where A = DP, FL, or SW. We use the Laplace transform (LT) technique (e.g. Carslaw and Jaeger [58]) to solve (E.1) for each layer. Applying this transform to (E.1), we obtain V20A
—
[qA(s)]20A=
~
s),
(E.2)
where 0( ~, s) is the LT of ø( ~, T), s is the LT variable, q~(s)as \/~~ and 4’A(~,
s) as PeA[DaA~’A(~,s) +
19A(~,
which contains the ND initial conditions
0)], 19A(
~, 0). The general solution of (E. 1) is given by
(E.3)
402
B. Svendsen et a!., Optical radiationfrom shock-compressed materials and interfaces
0A(~,s)=bA(s)e~+ cA(s)e_~+ 0AP(~,s),
(E.4)
with
0AP(~,s)=~
f~e~_~—e_’~)~A(~,s)d~
being a particular solution of (E.2). The boundary conditions include the requirements that 19,.( ~ —cc and +cc, respectively. Consequently, we have and 19~(~, ‘r) remain bounded as ~
T)
~
0
cD
as—
“~
J
~)
‘t’D\b’ I
b
b~=
q~0 Using these and writing (E.4) for each layer, we have ~
s) as bD
~~Df+
s),
0Dp(~’
bFe~ + cFe
0,.(4,s)=
+
—cc< ~
(E.6)
0~
(E.7)
are now given by
~J J
0DP(~,s)=
0~(~, s)
=
(E.5)
0FP(~,s), —l~~
~ where 0,~ and ~
~ —l ,
~
e
~Je’~D(~,s)d~,
~D(~,s)d~+
e~’~~( ~, s) d~+ ~
[e~~w(~,
(E.8)
s) d~.
(E.9)
Relations (E.5)—(E.7) contain four unknowns, bD, bF and cF, and c~, requiring us to specify four more boundary conditions. We obtain these by assuming the usual continuity of energy flux and temperature across each interface in the target (appendix A.2). In the transform domain, these are given by —kD ATD
(l/s)TD
+
8f0~J(—l,s) = —kF ATF a~0, ATD OD(—l, s)
—kF ATF dfOF(O, s)
(1/s)TF
+
as
=
(l/s)TF
+
1(—1,s),
(E.10)
ATF 052(1, s),
(E.11)
—k~AT~c9f0w(O, s),
AT,, OF(0, s) as (1/s)T~+ AT~0~(0,s)
(E.12) .
(E.13)
B. Svendsen cc a!., Optical radiation from shock-compressed materials and interfaces
403
Putting (E.6)—(E.9) into (E.10)—(E.13), we obtain
II~DFe e~ 0
I
~ 1 1 —i ~
0
b
as ODFATD
~(
~~F 1 1
0 \ /ATD ATF bF bD\
I/b,\ b
2 O~WFI~ ATFcF J1b31’
(E.14)
—1/ \AT~c~/ \b4/
j- eD(~,s)d~, —1
r eD(~,s)d~,
(7’,.— TD)— ~
b2as
2q~
f
.1
a
0
b 3~~ as ATF
1
(e~’+e~F~)~(~, s) d~+ ~WFATW ~ 2q~ .1
s) d~, a
0
b4as~(T~_TF)—~-~ ~
2q~ .1
2q~0
Assuming the coefficient matrix is nonsingular, i.e., 1 + a~DF)(e e Li(s) = —(1 + ~WF)( where
)$0,
—
(qD~qF)
1/2
~WF
1/2 as (kDpDcPDIkFpFcPF)
(kWpWcPW/kFpFcPF)
~
IIDF_(O•DF1)/(tTDF+1),
the system (E. 14) has a unique solution. Noting that 1
Li(s)
=
(1
+ ~DF)(1 + ~WF)(1
—
1-~.JF~WF
e qD =
(1
+ ~DF)(1 + ~wF) ~
~
~WF)
~_~F(2m+t)
, for —1< ~y~ 52
~
1
we invert the coefficient matrix in (E. 14) to obtain e~(I
/ATD bD\ (ATFbF
I\ATW ATF ~ C~/ cF
=
a,, (l—rwF)
a,2 (oWF—l)rD,.
(1 + ~WF)UDF —(1 2
+ ~WF)~DF
2~DF
(1+UDF) (1
—
0DF)
2o~,,e~ \ /b,\ (1+~DF)rWFe~
e~’ (1
—
ODF)OWF ~
a 43
a44
I
b 3/I’
I \b4~ (E.15)
B.
404
1” + ~WFe_~)~
a11
a
43
Svendsen et aL, Optical radiation from shock-compressed materials and interfaces
_
(1
+
(1
+ oDF)(e
(TWF)(e
—
~WFe~)
—
p
~
,
e~F),
a,2
as
(1
a44
as
—(1
+
o-WF)(e’ + a.DF)(e~17+
e~F)
ii~
Expanding these and substituting them into (E.5)—(E.7), we obtain the solutions in the LT domain. We then transform these back to the time domain using the convolution theorem. For the calculations presented in figs. 5.2—5.5, we require the temperature field for —1~~ ~0 and 0 ~ and ‘r>O, i.e. in the FL and SW. For the FL, we obtain
J
—,
T(~,r)=TF+ATD
T
JY~(~, ~
~)d~d~+B(~,r)(TD— TF)
—a o
ff OT
+
ATF
G,,(~,~,r
d~d~+ E(~,T)(TF
~)
— ~)~F(~’
—
T~)
—1 0
a +
1
ATwJ~ j ~‘F~ ‘~,~,r
~
—
d~d~,
~)
(E.16)
00
~
~,T)
1DF / 1 \1/2 ~ ~DF IT peDT) 1+
as
—
B(~,r) as 1
G,,(~, ~,T)as
pw,, exp{—[(2m ‘~
~DF
~
+
+
a
~
m=O
1)
(v,.~,,~~,.)m[exp{—[(2m + 1)
—
—
1/
—
KFD(l + ~)]2w2}
KFD(l +
m
(~DF~wF)[erfc{[(2m
+
~DF Pn0
2
+~
1)
+
~]wF} v~erfc{[(2m —
+
1)— ~]wF}], (E.17)
i
1/2
~ITPeFT~
a
~
~
m=0
/
rexp{—[2m+(~—~)]2w~}— v~,,exp{—[2m—(~+fl]2w~} -yDFexp{-[2(m
+
1) +
(~+ ~)]2w2}
I +vDFvWFexp{[2(m+1)(~fl]wF} ~ x~ 1 exp{—[2m + (~~_ P~~exp{—[2m (~ + ~)]2~}
—
—
-vDFexp{-[2(m+l)+(~+~)]wF}
~~0,
+~FvWFexp{[2(m+1)(~~)]wF}, ~
11 ~,T)as ‘F~S~
1 +UWF ~WF /
1
\1/2
~
a
(vz~~°
m=0
x
[exp{—(2m~ + K,.wfl2w~} —
—
ii~
exp{—[2(m
+
1) +
4 + KFwfl2W~}],
B. E(~, r) as
1
Similarly, for 0
(7
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
erfc{[2(m +1) ~ (~F~wF)m[~F ~WF m0
+WF
~
+
~]w,,} erfc[(2m —
—
405
~)wF]].
(E.18)
i.e. the SW, we obtain
—1
T(~,T)=Tw+ATDf ~ 0 Or
+ATF J~ J —w ~ (~, ~ —1
~)d~d~+R(~,r)(T,,—
T~)
0
aT
+ATwffGw(~,~,T
~)d~ d~,
~
(E.19)
00
1 1/2 (irpeDr) ~
2TDF ~
~,r)as
(1 + (7DF)(1 +
~
x exp{—[(2m + 1) +
Q(~,r) as (1+ (7DF)(1 1
+ ~WF) ~
K,.w~— KFD(1 +
(~DF~wF)m erfc{[(2m
1/2
1”
/
1/2
~
—
~DFexp{—[2(m
+1) + ~ +
a
~ (~DF~wF)m
m=O
rexp{—[2m
I I
a
z.~ (~DF~WF)
—
1 / 1 ~ ~irPe~r/
1) +
maO
x [exp{—(2m ~+ ~,r)
+
‘~
—w” ~ ‘~,~,r)as~)
~
~
+
KFw(4~—~)]2w~) + ~WFexp{—[2m
~,.exp{—2(m
+
1)
+ K,.w(~+
+ KFW(~+ ~)]22}
—~FvWFexp{—[2(m+1)+KFW(~—~)]wF},
x~ I exp{—[2m ~DF
+ K,.w(~—
exp{—[2(m
+
~)]2w~}+ ~WFexp{—[2m 1) +
+ KFw(~+
~)I2w2}
KFW(~+
—vDFpWFexp{—[2(m+1)+KFW(~—~)]2w~}, 4~~cc,
R(e, T) as 1+1
~
(VDFPWF)m[erfc{(2m
~WF m0
Each of the source terms now has the form
+ KFw~)WF) +
~,.
erfc{[2(m
+1) + KFw~]W~)].
406
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
PeA[DaA~’A(~
~A( ~, T) as
r) + 19( ~, 0)~(r)],
(E.20)
where 6(i-) is the Dirac delta function. As discussed in the text, we assume that compression/release processes in the target establish the initial (t as 0) temperature profile (5.2) in the DP—FL—SW system. Assuming ~D( ~, 0) = 0, ~( ~, 0) = 0 and ~‘F(~’°)—O’we have ~w(~,T)O, 4D(~’‘r)
=
PeD[h(~ + 1 + 6~) h(~+ 1)] ATDF/ATD, AT —
AT
(E.21)
~ ATF +PeF[h(~+~FW) h(fl] ATF from (5.2) and (E.20) written for the DP and FL, respectively, where h(x) is the unit-step function, and
6~’as61/d.Putting (E.21) into (E.16), we obtain the temperature in the FL (—1~s0, r>0), ~ +
E(~,T)(TF
—
(E.22)
T~), m[erfc{[(2m
A(~,T)as 1 —
DF
m=O ~
+
1)+ ~]w,,) —erfc{[(2m
+
1)+
~ + KFD8~]wF}
(VDFVWF)
v~,,(erfc{[(2m + 1)— ~]w,,}—erfc{[(2m+ 1)— ~ + KFD3~,]WF})],
C(~,r)asg~(~,r, —1, ~)+g~(~,T,
—l+6~),
4,
~
(E.24)
~1+ôFD,
(E.25)
~
~ g~, r, a, b) as
~
(~~FvWF)m[±erfc{[2m ± (a
+
vwF(erfc{[2m
—
PDF(erfc{[2(m
i
—
+
(~ +
a)]w,,}
—
—
~)]w,.} ~
erfc{[2m
—
erfc{[2m ±(b
(~ +
+
1) ~ (a
—
~)]w,,} erfc{[2(m —
—
b)]w~})
1) +(~+ a)]w,,} —erfc{[2(m+ 1)
VDFVWF(erfc{[2(m
(E.23)
+ (~+
+
1)
~
b)]WF})
(b
—
(E.26) The function g~( ~, T, a, b) results from the integration of the second term on the right-hand side of (E. 16) using (E .21); a and b are the lower and upper limits of the spatial integration, respectively. We obtain g~from g~by using the upper signs in each term having both + and minus signs, while g~is given by the lower signs in each term. Lastly, with 4w( ~, r) as 0 in (E. 19), we obtain the temperature in the SW (0~4, r>0), —
B.
407
Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
T(~,r) Tw+A(~,r)ATDF+Q(~,r)(TD—TF)+Q(~,r,—1,—1+6D)ATFD +
fi(~,r, —o~,0)AT,,~ + R(~,‘r)(TF ~DF
—
Il
\I1
x [erfc{[(2m r, a, b)
V m \ ~ (~DF~WF) °WFI m=O
+
1)
+ K,,w~+
KFD(~l~]WF}
m[erfc{(2m 1~WF m—0 ~ (t/DFPWF)
1
as
L
(E.27)
T~),
+ VDF(erfc{[2(m +
1)
+ KFW~+
+ erfc{[(2m +
+
1)
KFw~ b)WF}
b]w,,}
—
—
erfc{[2(m
+
—
erfc{(2m
+
1)
+
K,,w~ a)wF} —
+ K,,w~+ a]WF})].
Relations (E.22) and (E.27) are those used in the text to examine the effect of conduction on the temperature field around the FL—SW interface.
Appendix F. Radiative transport model As discussed briefly in the text, radiation from sources at the FL—SW interface and/or the SW may be affected by radiative transfer through the SW, shock front, unshocked TW and its free surface. In general, radiation sources include (1) the FL at the interface, (2) gas or other trapped material at the interface and/or (3) the SW. The FL is generally a metal (i.e. an opaque material), while the TW is generally an oxide, halide or silicate. Many of these latter materials are of natural origin, possessing color and containing inclusions of various sizes, some of which are potential scatterers (e.g. apparent Rayleigh scattering in CaO: Boslough et al. [72]). These possibilities motivate us to formulate a model from (D.3) and the geometry of the target of sufficient generality to deal with emission, absorption and! or isotropic scattering in the shocked and/or unshocked TW. Note that in writing (D.3), we have already assumed isotropic scattering. For tractability, we must accept this as a limitation of the model. We have no a priori reason, of course, to expect this to be true in shocked or unshocked TW material containing scatterers. Since our experiments are calibrated, we need worry only about the differences between calibration and experimental configurations, which include radiation sources, and the effects of the (1) SW, (2) shock front, (3) USW, and (4) TW free surface, on the source radiation. For simplicity, we assume the target may be approximated optically as a set of parallel plane layers. In this representation, the temperature and material properties of the FL, SW and USW are assumed to vary at most only along the direction of shock propagation, x. Further, we assume that i~is axisymmetric about s; in this case, i~ depends only on s = s~and the angle between s and x, which we designate 4, so that i~= i~(s,~5). Introducing the extinction coefficient, K 5 as a5 + (75, the differential opacity, d K5 as K5 (s) ds, and the albedo for scattering, ~ as o-5/K5, into (D.3), with ds as dx/cos 4, the radiation intensity in all directions forward (‘—b +) and backward (‘—s. —) in a plane layer is given by the solution of (e.g. Siegel and Howell [15]) /_).+I
\\
( at5 ~K5, /L, I ) ilK5
.+
~ /2
.,
I-)
t~
~so
I
.+
[15(K5,s
*
.—
)+z5(K5,—~
*
)]dp.
*
,
L 0
(F.1)
408
B. Svendsen et al., Optical radiation from shock-compressed materials and interfaces
) aK5
+
z,,, (K5,
—p~)=
(1—
+
In these relations,
(Iso) lse( K5,
—~)
/2
~
f
[i~5,
~*)
+
i~(K5,_~*)] d~*.
(F.2)
j~as C05
(F.3)
Ks(x)asJK5(~)d~
is now a function of x. Radiative transport in each layer of the target is then governed by (F. 1) and (F.2). Given a target geometry like that of figs. 2.1 and 2.2, we assume that observed radiation is dominated by radiation from the target of nearly normal incidence (i.e., ~ as 1). This is the fundamental simplifying assumption of the radiative transport model used in our work. Mathematically this assumption takes the form i~(K5,~
as i~(K5)ô(1 —
p~,
(F.4)
where 5(x) is Dirac’s delta function. In effect, this assumption imposes a very special averaging on the optical properties of the model, i.e., that model properties are dominated by incident radiation perpendicular to the layering of the target. Putting (F.4) into (F.1) and (F.2), we obtain di~/dK5+ i~=
(1
—
—di~1dK5+ i~= (1
—
+
as
~Q50(i~ + i~), +
~so)~e(”s)
respectively. Now let ~ as subtracting them, we obtain
dI5JdK5
+
~so)~e(’~s)
±~
(F.5)
(F.6)
~Q50(i~ + i~),
Substituting 1~÷ and I~ into (F.5) and (F.6), adding and
2I3~~,~e(Ks) + ~so’s÷ ‘
(F.7)
d15~/dK5+15_=0, with
\/i
$~=
—
(F.8)
-
Q~.Eliminating I~ from these expressions gives us a differential equation for I~ i.e., +~
21 d
5÷/dK~f3~I~= ~2/3~~,~e(Ks).
(F.9)
—
For ~ ~ some f(K5), which is the case if we assume that a5Ia-5 ~f(K5) general solution of (F.9) is given by
I5~(K5)= C e~
+
D e~
+
J
in each layer of the model, the
sinh{~5(K~ K5)) ~~e(K~) dK~. —
(F.10)
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
409
Then (F.8) provides
J
1 5(K5) =
—d15~!dK5as —~5Ce~
+
~5D e~
+
~
cosh{~5(K~ K5)} —
1~e(K~) dK~.
(F.ll)
Combining (F.10), (F.11) and the definitions of 1~±, gives us i~(K5)= ~
± J~)=
J
i~(K5)= ~
~(1 ~ ~5)Ce~
[(1 ±$~) ~
~(1 ± f35)D ~
+
—(1 ~
+
$~)e~]i~~(K~)
(F.12)
i~(K5),
dK~.
From (F. 12), we have the following conditions at the boundaries of the layer for backward and forward propagating radiation: ~(1+/35)C+
i(0)=
i~(KSd)= ~(l
+
~(1—f35)D,
— /35)D
~
+
$5)Ce~”~’~ + ~(1 + f35)D
~
+ i~P(KAd)
f35)C e~”~ + ~(l
i(0)= ~(1—f35)C+ i~(KSd)= ~(1
—
~(1+/35)D
where K5,~= K5(d), with d the layer thickness. We need only two of these to find C and D; for reasons apparent later, we choose the second and third relations. Solving these for C and D, and substituting the resulting expressions into the first and fourth relations, we obtain
(
i~ 0
—
2(1
4PS[i~(K e~2d 5d) i~P(Ksd)] ,
13~)(1 sinh(f3SKSd) + e~2d i~(0)(1+ —
—
~)2
—
~)2
4f35i,~(0)+ 2(1 f3~)sinh(I3SKAd) [i~(KAd) i,~P(KAd)] (1 + ~)2 e/~K20j—(1 e~~’~’°” + 1SP(KAd) —
t~(,ç~) =
( F . 13 )
—
—
—
~)2
(F.14)
.
—
With f3~=
~ and KSd = Kss,., these relations apply to the SW; with f3~= $susw’ Ksd = K5FS K5SF and 0 (no sources), they apply to the unshocked TW. Next, we assume that (1) the opacity of the FL is sufficiently large so that any radiation observed from it originates near the interface, and (2) all boundaries are optically smooth, so that radiation incident on these boundaries is refracted and reflected according to Fresnel’s laws (e.g. Siegel and Howell [15]). Basically, by doing this, we neglect any scattering properties these boundaries may have. Using (F.13) and (F.14), and assigning each boundary of the target an intrinsic normal spectral reflectivity, r5, given by (Siegel and Howell [15])
i~P(KSd)=
2 + (w5A rAAB as [(~5A
—
~AB)
—
~5B)]’[(~5A
+ ~5B)
+ (w5A +
WAB)]
410
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
for normal incidence on the boundary between layers A and B possessing complex indices of refraction t05A and ~5B + la/SB (where w ~sA + ‘ 5 ~ a5), respectively, we may construct the boundary and layer conditions for the model. At the interface (x = K5 = 0), radiation traveling forward into the SW, i ~~(0)~ is composed of radiation emerging at the SW side of the interface from the FL and! or gas, S ~(0), plus that component of the backward traveling radiation, i ~~(O), reflected off the interface back into the forward direction, i.e., i~~~(0) = S~1(0)+ r5ii~~~(0).
(F.15)
This is the boundary condition at the interface. From (F.13) and (F.14), we have conditions for backward and forward propagating radiation in the SW [0~ x ~ xx,,, 0< K5 < KSSF = KS(xsF)], i.e., ~sw(0)
=
c1i~~~(0) + c2i~w(K5s,,)+ S~w(KssF), c2i~~(0) + c1i~~(K55,,) + S~sw(KssF),
j~sw(KssF)=
respectively.
(F.16)
At the shock front [x = XSF, KSSF
=
(F.17)
KA(xsF)], radiation is transmitted and reflected in both
directions. Consequently,
(F.18)
i~sw(K5s,.) = c3IAUSW(KSSF) + T5SFIASW(KSSF) ,
~~usw(’AsF) = c4iAsw(KAsF) + rASFLAUSW(KSSF)
(F.19)
.
Again using (F.13) and (F.14), the conditions in the unshocked window KAFS = KA(XFS)] are given by
[XSF
~ x ~ XFS, KSSF ~ K5 ~
iusw(K5s,,) = c,iUsw(KASF) + c6i~~Usw(KAFs) ,
(F.20)
j~~~k552~ ) = c6iusw(KAsF) + cSZAUSW(kAFS) ,
(F. 21)
where we have assumed ~e~~(K~) as 0, i.e. no emission from the unshocked window. Lastly, at the unshocked window free surface [x = XFS, KSFS as KS(xFs)], radiation traveling back into the unshocked window, iusw(KSFs)’ is equal to that component of the forward traveling radiation incident at the free surface, i~usw(Ks,,s) [x = XFS, KSFS = K5(x,,5)], reflected into the window, i.e., i~USW(KAFS)= rAFSZAUSW(KAFS)
(F.22)
.
The coefficients and source terms in these relations are given by
—
c,= 2(1
—
~
Asinh(J3SSWKASF)
4I3~sw
fl
,
3—i Asw
c
=
5—
c
,
2(1
—~
Ssw 413S~w
sinh[f35Usw(KSFS
—
K5SF)]
=
, AUsw
c6_,~ 5Usw
rSsFasc4,
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces
S
+
—
.+
(1
51(0)
—
rsj)lsei(O)
35sw
/
—
A
~ssw(1(ssF) as
‘
411
l5sWP(KASF)
55W
2(1
.÷
÷
S5s~(K5s52)as 1SSWP(KSSF)
—
~
sinh(/355~K5552)
A
—
5
lSsWP(KASF)
SW
where (1 + Asusw
=
/3)2
ePAsw”¾5b2—
~l + t-’susw) ~
~2
‘.
(1
—
e~1\~55F
$)2
~
—
\
—
eu1F~~~sb2)
~ ~ t-’sUSWI
Relations (F.15)—(F.22) constitute 8 equations in 8 unknowns. We are particularly interested in calculating i~usw(K552s),since the radiation escaping the target destined for detection is then given by tsmod(KSFS) as
(1
— rSFS)IAUSW(KSFS)
(F.23)
.
The system (F.15)—(F.22) may be cast into the following matrix form: S~~(0)
—r51 1 0 0 0 0 0 0
0
0
0
0
0
—c1—c2
1
0
0
0
0
0
0
_~2
~
_~,
0 0 0 0 0
1 0 0 0 0
i~5~(0) .-
~
rSSF
c4
c3
rASF
0 0 0
1 0 0
+
155w(K5SF)
0 1 —c, c6 0
0 0 —c6 —c5 1
0 0 0 1
S~5~(K5552) S5SW(K5SF)
1ASW@~ASF)
i~ (K
~
0 0
5SF
t5usw(KASF) i ~‘K~
rFS
0 0
SUSW\ SFSI
i~usw(K552s)
0 (F.24)
In matrix notation, this is [A]{i} = {d}, where [A] is a reflectivity—transmissivity matrix, {i} is a vector of the forward and backward traveling radiation intensities at the boundaries, and { d } is a radiation source vector. Decomposing [A] into upper and lower bidiagonal matrices, [A ~] and [AL], respectively, we find the solution of [AL]{u) = {d} and use this to solve {i} as [A~]’{v}. Solving this system gives us an expression for iU5W(KAF5), which we substitute into (F.22) to obtain jSfflOd(KSFS)
as
~P~[S51(0)+ SSSW(KSSF)]
(F.25)
.
In this relation, ~ is the effective transmissivity of the SW, shock front, unshocked TW and its free surface, given by =
E~(1 r55F)(l
as
[(1
—
— r51c1)(l
—[(1
—
—
—
r5,,5)c6 ,
c1r5552)
— r51c~r5~,,][(1
—
r51c1)c1 + r51c~](1
—
r5SF)
[(1
—
rASFc5)(l
c5rAFS)c5
—
cSrSFS)
+ c6rSFS]
—
rSSFc6rSFS]
412
~
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces is
the intensity of sources in the SW along K5, i.e., A SF
f
SASW(KSSF)as
A5(K55~, ~
0
1 As(KssF,
~) as
~
3~~~) [c, + (c~
r
—
/3ssW{(
—
51c1)(1
+
— ~/3~~~{(1— r51c,)(1
—
I —
c~)r 51](1— ~3~~)} ew~F)
$~~) —
[c1 + (c~ c~)r51](1+ /35sW)} e_w_K25~, —
and S51(0) is the combined effective intensity of all interface sources, S51(0)
c2(1
as
— rsj)i~ei(0)
Relation (F.25) is the principal result of this appendix, and constitutes the simplest model incorporating scattering, absorption and multiple reflections that we can derive for radiative transport in the target. As it stands, (F.25) is sufficiently general for comparison with observations from a number of different radiation experiments. At this point, however, our main interest is in the interface experiments discussed in the text, so we now specialize (F.25) to this purpose. For cases3SUSW where= there is apparently scattering in the SW, and none in the unshocked TW, we may 1 in (F.25); in thisno case, c set ~ = / 1 = c5 = 0, c2
as =
c6
as e’~5F
(1
— rSFSI-AUSWrASF)
AA(KSSF,
~) = rASW(e~+
as ~
=
—
TASW[rSFS(l
(F.26)
,
—
2r55,,)c~+
(F.27)
rSSF]rAI ,
(F.28)
r51 e~) .
From (F.27), we note that, in general, neglect of multiple reflections when {r5, r5}—* 1 is clearly incorrect and can lead to underestimate of the source intensity by the model. However, for the experiments of interest here, we do have r5 0. In this case, 2rASF)rSI = 1 0(r2), (F.29) H5(A, t) ~ 1 — (r5FS + rAI)rSSF — rAFs(1 and multiple reflections are, to first order, negligible. Now if we assume each region of the dynamic target optical geometry possesses a distinct extinction coefficient, and that, in each of these regions, —
—
—
optical properties are homogeneous, we have
K 55~, K5(x)= KSSF, ~
O
(F.30)
XSF
where x~,,is the position of the shock front in the SW, x~Fits position in the USW, and
4,,
—~
is
the
shock front thickness. Putting (F.30) into (F.3), we obtain K55,,(X) = KASWX~F,
(F.31)
B. Svendsen et a!., Optical radiation from shock-compressed materials and interfaces KAFS(X) = K
55,,(x) +
K5~,,(x~52xsF) —
+
413
(F.32)
K5~5~(dx~F). —
For steady shock wave propagation, x and t are not independent, and we may write x~,,(t)= (u
—
x~F(t)= (u
U)t,
u~)t,
—
(F.33)
where u is the speed of displacement of the shock front, and u - and u are the material velocities behind and in front of the shock front (notation of appendix A.1). Putting these into (F.31) and (F.32), we have +
KSSF(t)
=
KSSW(u
KAFS(t)
as KASF(t)
—
(F.34)
u)t,
+ K5sF(U
—
v~)t+ K5~~~[d(u —
—
u~)t].
(F.35)
Defining as
~
K55~(u
—
= KSSF(U as
—
U)texp,
(F.36)
U~)texp‘
(F.37) (F.38)
~
as nondimensional extinction coefficients, and setting u —
j,’*
as
0, (F.34) and (F.35) may be written
~
(17
3
ASW ‘‘exp ,
—
KSFS(t)
+
KS5F(t) + K~SFt/texp+
K~’~~~(1 t/texp) ,
(F.40)
—
respectively. If we treat the shock front as a 2D boundary, rather than a thin layer (appendix A. 1) K55,,—~O.In this case, (F.40) simplifies to KSFS(t) = K5s,,(t) +
K~’UsW(1
—
t/texp)
(F.41)
.
Relations (F.39) and (F.41) are the expression we use in (F.26) and the text. Lastly, if we assume that sources in the SW are distributed uniformly, ~ ~) is independent of x, i.e. spatially uniform, and from (F.22) this implies KASF
SASW(KSSF)
assuming that
as
f
i.~eSWis
AA(KSSF,
~)~~esw(~)
d~as (1
—
r5~~)(l + rSIr5SW)IA+(A, T~),
(F.42)
given by the Planck function. Putting these results into (F.25) and (F.26), we
obtain, with S~= 1, tAmod(A, t) as êssW(t)ISpl(A,
T~)+ â51(t)I5~1(A,T1(t))
,
(F.43)
414
B. Svendsen ci al., Optical radiation from shock-compressed materials and interfaces
where
e5~~(t) = ~1~(1
—
TASW)(l +
r51’r55~),
ê51(t) as ~P~TAsW(l
—
r51)
are the effective normal spectral emissivities of the SW and FL at the FL—SW interface, and as (1 — rAAB)T5USW(l — r5SF) with E,~= 1. Relation (F.43) is the expression we use in the text, with tAmod~
The Hugoniot temperature of the SW, T~,is assumed homogeneous, uniform, and
constant with a uniform distribution of SW sources. The interface temperature, T,(t), however, may be ‘Amod a function of time, or constant, in the context of the conduction model discussed in appendix E.
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