Corresponding author: R. G. Kraus, Department of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138, USA. (firstname.lastname@example.org)
 The most energetic planetary collisions attain shock pressures that result in abundant melting and vaporization. Accurate predictions of the extent of melting and vaporization require knowledge of vast regions of the phase diagrams of the constituent materials. To reach the liquid-vapor phase boundary of silica, we conducted uniaxial shock-and-release experiments, where quartz was shocked to a state sufficient to initiate vaporization upon isentropic decompression (hundreds of GPa). The apparent temperature of the decompressing fluid was measured with a streaked optical pyrometer, and the bulk density was inferred by stagnation onto a standard window. To interpret the observed post-shock temperatures, we developed a model for the apparent temperature of a material isentropically decompressing through the liquid-vapor coexistence region. Using published thermodynamic data, we revised the liquid-vapor boundary for silica and calculated the entropy on the quartz Hugoniot. The silica post-shock temperature measurements, up to entropies beyond the critical point, are in excellent qualitative agreement with the predictions from the decompressing two-phase mixture model. Shock-and-release experiments provide an accurate measurement of the temperature on the phase boundary for entropies below the critical point, with increasing uncertainties near and above the critical point entropy. Our new criteria for shock-induced vaporization of quartz are much lower than previous estimates, primarily because of the revised entropy on the Hugoniot. As the thermodynamics of other silicates are expected to be similar to quartz, vaporization is a significant process during high-velocity planetary collisions.
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 During the end stage of planet formation, the nebular gas disperses and mutual encounter velocities increase via gravitational stirring from the largest bodies. N-body simulations of this stage find typical collision velocities between protoplanets of one to a few times the two-body escape velocity [Agnor et al., 1999]. The kinetic energy of an impact is partially transferred to internal energy in the colliding bodies via passage of a strong shock wave. At the expected impact velocities of ∼10 to a few tens of km s−1onto the growing planets, the internal energy increase is sufficient to melt and vaporize a large fraction of the colliding bodies. However, the predicted degree of melting and vaporization for a specific impact scenario has great uncertainty that primarily arises from poorly-constrained equations of state (EOS).
 Accurate equations of state over a tremendous region of phase space are required to make predictions about the impact processes so prevalent during the formation of the solar system and its subsequent evolution. The last giant impact is invoked to explain the diverse characteristics of rocky and icy planets in the Solar System [e.g., Stewart and Leinhardt, 2012], including the large core of Mercury [Benz et al., 1988, 2007], formation of Earth's moon [Canup and Asphaug, 2001], Pluto's moons [Canup, 2005], and Haumea's moons and family members [Leinhardt et al., 2010]. In each case, the argument for a giant impact relies upon the details of the equations of state.
 Although equation of state theory is advancing rapidly, the generation of accurate and complete equations of state from first principles is still not feasible for most geologic materials. Planetary collisions are particularly challenging because of the need to understand both the extreme temperatures and high compression ratios achieved in the shocked states and the low densities and temperatures of shock-vaporized material. Experiments are required to constrain the equation of state throughout the phase space of interest.
 Planar shock wave experiments, typically on gas guns, have been used to define the shock states for a wide range of materials, including many planet-forming rocks and minerals [Ahrens and Johnson, 1995a, 1995b]. The range of states that are accessible by laboratory shock wave experiments has increased dramatically, to thousands of GPa, with the development of pulsed lasers [e.g., Hicks et al., 2005] and electromagnetically launched flyer plates [e.g., Knudson and Desjarlais, 2009]. The pressure, density, and internal energy increase can be accurately determined along the Hugoniot, the locus of shock states. For transparent materials, the shock temperature can also be determined by measuring the thermal emission from the shock front [Lyzenga and Ahrens, 1979; Miller et al., 2007].
 The states achieved during decompression from the shocked states are much more difficult to investigate due to the technical limitations in reaching this region of phase space. In particular, accessing the equation of state surface near the liquid-vapor curve and up to the critical point for most relevant Earth materials is technically difficult as the temperatures (2,000 to 10,000 K) and pressures (up to a few GPa) cannot be obtained through quasi-static laser-heating techniques. Resistive-heating techniques have been used to isobarically heat samples to the temperature range of interest [Gathers et al., 1976]; however, these techniques are only suitable for electrically conducting samples. Over the past 50 years, a significant number of shock-and-release experiments have been performed to probe the pressures, and in some cases the temperatures, along the liquid-vapor curve of the more volatile metals: lead [Ageev et al., 1988; Pyalling et al., 1998], tin [Bakanova et al., 1983; Ternovoi et al., 1998], bismuth [Kvitov et al., 1991], and cadmium [Bakanova et al., 1983]. In these experiments, the pressure at the intersection between the release isentrope and liquid-vapor curve was determined by measuring the shock generated in a downrange helium gas cell. Usually, the helium gas backing the metal was shocked to temperatures much higher than the release temperature in the metal, severely complicating the simultaneous study of the thermal emission from the metal.
 Free-surface post-shock pyrometry measurements were performed on aluminum and cerium shocked to multiMbar pressures [Celliers and Ng, 1993; Reinhart et al., 2008]; however, no other thermodynamic parameters were measured or associated with the apparent post-shock temperatures.Stewart et al. measured the shock and apparent free-surface post-shock temperatures for H2O ice shocked into the fluid phase. Using a model for the entropy along the Hugoniot and assuming high emissivity, the post-shock states fell on the known liquid-vapor curve for water.
 Recently, Kurosawa et al. used laser-launched flyers to shock diopside and quartz into a fluid state. Upon breakout of the decaying shock wave at the downrange free surface, a spectrograph coupled to an ICCD was used to observe late time emission lines from atomic and ionic species. Using a model for pressure broadening, the authors were able to estimate the temperature and pressure in the expanding vapor plume. However, the decaying-shock measurements did not constrain the liquid-vapor phase boundary for these materials.
 In this study, we focus on silica, one of the most abundant minerals in the Earth's crust and an end-member composition of the mantle of rocky bodies. Throughout this study, quartz refers to the crystallineα-quartz phase of silica and fused quartz is explicitly noted. Variable definitions and annotations are defined inTable 1.
Table 1. Summary of Primary Variables and Annotations
General Thermodynamic Variables and Constants
Ideal gas constant
Specific internal energy
Specific Gibbs free energy
Thermodynamic Grüneisen parameter
Exponent to volume dependence of Grüneisen parameter
Isochoric heat capacity
Isobaric heat capacity
Subscript denoting critical point value
Absolute Entropy Calculation Variables (§2.1)
Subscript denoting reference state of stishovite at STP
Subscript denoting intermediate state: stishovite on the principal isentrope at VB
Subscript denoting intermediate state: stishovite on the melting curve at 4500 K
Subscript denoting intermediate state: fluid silica at intersection of fused quartz Hugoniot and melting curve
Subscript denoting reference state of fluid silica at intersection of quartz Hugoniot and melting curve
Subscript denoting state along the quartz Hugoniot
Subscript denoting state along the fused quartz Hugoniot
Entropy of isochorically heating stishovite from TA to 4500 K
Entropy of melting stishovite at 4500 K
Absolute uncertainty in the specific entropy
Absolute uncertainty in the temperature
Stagnation Variables (§3.2)
Drive laser intensity
Impact velocity of released silica onto LiF window
Average density of decompressing fluid immediately prior to impact with LiF window
Shock pressure induced in LiF window upon stagnation
Shock velocity in re-shocked silica fluid
Particle velocity in re-shocked silica fluid
Velocity of interface between stagnating silica and LiF window
Intercept of US,SF − uP,SF relation
Slope of US,SF − uP,SF relation
Phase Decomposition Variables (§5.1)
Timescale for dynamic fragmentation to occur within the metastable fluid
Adiabatic sound speed
Fragmentation length scale
Timescale for nucleation and growth to occur within the metastable fluid
Kinetic coefficient for nucleation
Number density of particles in volume V*
Volume undergoing homogeneous nucleation
Work to form a critical size nucleus
Pressure difference between metastable state and liquid–vapor curve
Post-shock Temperature Buffering Variables (§5.2)
Specific entropy of the liquid-vapor mixture
Temperature on liquid-vapor curve at specific entropySm
Average specific volume of liquid-vapor mixture
Mass fraction of vapor
Slope of the liquid-vapor curve at temperatureT
Subscript denoting the liquid phase
Subscript denoting the vapor phase
Post-shock Density Profile Variables (§5.3)
Density along characteristic i
Eulerian sound speed at densityρi
Initial density in Lagrangian coordinates
Radiative Transfer Model Variables (§5.4)
Apparent post-shock temperature
Probability of a photon being absorbed at position x within a slice of thickness dx
Average absorption coefficient
Mean free path of a photon interacting with a liquid droplet
Absorption coefficient of the vapor phase
Liquid droplet diameter
Eulerian position of free surface of releasing fluid
Eulerian position of the emitting region within the liquid-vapor mixture
 In section 2, we discuss the thermodynamics of shock compression and release, with particular emphasis on understanding entropy. We calculate entropy at new reference states to determine the absolute entropy along the quartz principal Hugoniot, and the boiling point of silica is recalculated using recent thermodynamic data for liquid silica. Then, we discuss the experimental techniques used in our new shock-and-release experiments that probe the temperature and density along the liquid-vapor curve up to the critical point (section 3). In section 4, we compare the results of our post-shock temperature measurements with a revised model for the liquid-vapor curve of silica. Next, we consider the timescales for phase decomposition during decompression (section 5.1), which we find are short compared to the timescale of the experiments. To aid in interpretation of the experiments, we calculate the temperature and density profiles along release isentropes and as a function of distance in the expanding post-shock mixture (sections 5.2–5.3). Then, we develop a radiative transfer model to predict the apparent post-shock temperature of the two-phase mixture as a function of entropy (section 5.4). The predicted post-shock temperatures and the measured post-shock temperatures are compared insection 6.1. The agreement between the radiative transfer model and the experimental data indicates that the shock-and-release technique provides robust constraints on the liquid-vapor curve. Finally, we calculate the critical shock pressures for shock-induced vaporization (section 6.2) and discuss the implications for planetary collisions (section 6.3).
2. Reference Entropy States for Silica
 The shock Hugoniot follows a complicated path through the EOS surface that depends on the equilibrium EOS, material strength, and kinetics related to phase transitions. Across the shock front mass, momentum, and energy are conserved, which allows one to exactly solve for the increase in internal energy, longitudinal stress, and density in the shocked state using the Rankine-Hugoniot equations [Rice et al., 1958]. Shock pressures above ∼110 GPa transform quartz to a fluid [Lyzenga et al., 1983; Hicks et al., 2006], where strength and kinetics of phase transformations have no effect on the path of the Hugoniot through the EOS surface. Figure 1presents the pressure-temperature phase diagram of silica with the Hugoniot for quartz in the fluid region as has been previously measured byHicks et al. . The solid-solid and solid-melt phase boundaries are based upon experimental data [Presnall, 1995; Akins and Ahrens, 2002, and references therein]. The liquid–vapor curve is from the M-ANEOS model for silica byMelosh .
 M-ANEOS is an equation of state formulation that is widely used in the planetary science community for modeling impact events. M-ANEOS [Melosh, 2007] is an updated version of the ANEOS model [Thompson and Lausen, 1972] that allows for the correct degrees of freedom available to molecular gases (vibrational, rotational, dissociation, ionization). In developing the M-ANEOS model for silica,Melosh constrained the liquid-vapor region by the thermodynamic data for the boiling point [Schick, 1960] and the triple point [Mysen and Kushiro, 1988]. The silica M-ANEOS model is currently lacking in some areas: there is no melting transition and molecular dissociation is only included in the gas phase. Yet, with only one phase transition (quartz to stishovite), the M-ANEOS model generally reproduces the silica shock velocity-particle velocity relation, hereafter referred to asUs − up, which ensures the internal energy gain along the Hugoniot is approximately correct. However, the temperatures along the M-ANEOS model Hugoniot are consistently much hotter than theHicks et al.  quartz shock temperature measurements, which were not available at the time of the model's development.
 The irreversible work from shock compression can be dissipated as either temperature or entropy (the − TS term in the free energy functions [e.g., Kittel and Kroemer, 1980]), where the balance between temperature and entropy depends upon the heat capacity. Hicks et al. found that the heat capacity in the fluid region on the quartz Hugoniot significantly exceeds the Dulong-Petit limit of 3kBper atom, which is the assumed limit in the M-ANEOS model. As a result, more irreversible work is partitioned to entropy than predicted by the M-ANEOS model.
 Although shock compression is irreversible, decompression from the shocked state is generally an isentropic process. Decompression will not be perfectly isentropic if the decompressing parcel of material becomes metastable upon going through a phase transition, has significant strength or viscosity, or conducts heat; however, these effects are generally negligible with respect to the entropy along the Hugoniot and can be neglected here. Consequently, for a parcel of material, the entropy in the decompressed state is effectively the same as the entropy in the shocked state. Using the P − T phase diagram (Figure 1), one needs an accurate model isentrope to predict the final state(s) of a material after release from the shocked state. However, by using an entropy-temperature (S − T) diagram, such as in Figure 2, one need only know the entropy along the Hugoniot and the entropies of the relevant phases at the final pressure in order to determine the final state(s). This realization is the basis for the “Entropy Method” [Ahrens and O'Keefe, 1972; Kieffer and Delany, 1979], which has been used in several studies to predict the amount of melting and vaporization that occurs during planetary impact events [e.g., Pierazzo et al., 1997; Artemieva and Lunine, 2003; Wünnemann et al., 2008; Kraus et al., 2011; Stewart et al., 2008]. Next, we present new calculations for reference entropy states for silica.
2.1. Entropy on the Quartz Hugoniot
 From the first law of thermodynamics, the change in entropy (dS) along any thermodynamic path is determined by the change in internal energy (dE) and specific volume (dV) at a given pressure (P) and temperature (T):
Although the Hugoniot is not the thermodynamic path taken by a material during the shock compression process, it is a valid thermodynamic path through the equation of state surface. On the quartz Hugoniot, the pressure, volume, and internal energy are known up to 1600 GPa with high accuracy (e.g., a few percent) [Knudson and Desjarlais, 2009], and the shock temperature is measured to 800 GPa with an uncertainty of less than 8% [Hicks et al., 2006]. The change in entropy along the Hugoniot may be obtained simply by integrating Equation (1); however, the shock temperatures in the solid phases are not sufficiently well-known to determine the absolute entropy by integrating from the standard state. Consequently, we used another thermodynamic path to determine the entropy at an alternate reference state: the point where the quartz Hugoniot enters the fluid region.
 The path to the fluid reference state on the quartz Hugoniot (state D in Tables 1 and 2) was calculated starting with stishovite in the standard state (T00 = 298 K, ρ00 = 4.209 g cm−3 [Weidner et al., 1982], S00 = 409 J kg−1 K−1 [Saxena, 1996; Mao et al., 2001]) via three intermediate states (A-C), whereS00 is defined relative to the perfect crystal at T = 0 K. Next, we present a calculation of the specific volume of the solid on the stishovite melting curve at the point crossed by the fused quartz Hugoniot.
Table 2. Summary of Thermodynamic States Calculated to Obtain the Entropy at the Intersection of the Quartz Hugoniot and the Melting Curve
Specific Volume [10−4 m3 kg−1]
Specific Entropy [J kg−1 K−1]
409 ± 20
Stishovite at STP
2.125 ± 0.027
409 ± 20
Stishovite on the principal isentrope at VB (334 K)
2.125 ± 0.027
3567 ± 54
Stishovite on the melting curve at 4500 K
2.153 ± 0.048
3820 ± 160
Fluid silica at intersection of fused quartz Hugoniot and melting curve (4500 K)
1.983 ± 0.029
3820 ± 168
Fluid silica at intersection of quartz Hugoniot and melting curve (4800 K)
 The fused quartz Hugoniot leaves the stishovite melting curve at 70 GPa and 4500 K [Luo and Ahrens, 2004]. The volume of stishovite at 70 GPa and ∼ 4100 K is 2.108(24) × 10−5 m3 kg−1 [Akins and Ahrens, 2002; Kraus et al., 2012], which was calculated from the quartz Hugoniot in the stishovite phase (Us = 944(308) + 1.99(0.10)up [m s−1], covariance = −30.69, fit to data presented in Marsh  and Trunin et al. ). Then, a thermal correction was applied to determine the specific volume of solid stishovite at 70 GPa and 4500 K: VA = 2.125(27) × 10−4 m3 kg. We used a coefficient of thermal expansion of α = 2 × 10−5 K−1, which is extrapolated from the value calculated by Luo et al.  at 120 GPa to 70 GPa by assuming the product of the coefficient of thermal expansion and isothermal compressibility [Panero et al., 2003] is a constant [Birch, 1952]. For a discussion on the validity of this approximation see Anderson .
 The temperature on the stishovite principal isentrope at VA was calculated via the thermodynamic relation,
where γ is the thermodynamic Grüneisen parameter. For the range of temperatures and volumes considered here, we assumed the Grüneisen parameter to be independent of temperature and follow an empirical function of volume,
where γ0 is the Grüneisen parameter at V00 = 1/ρ00 and q is a fitting parameter. For stishovite, γ0 = 1.35 [Watanabe, 1982] and q = 2.6 ± 0.2 [Luo et al., 2002]. Then, TA = 334 ± 4 K at VA.
 Next, we determine the entropy change upon isochorically heating stishovite from TA = 334 K to the melting curve, TB= 4500 K, using the experimentally-validated Kieffer model [Kieffer, 1979]. For the thermal properties of silicates, the Kieffer model is preferred over the Debye model as it includes the anisotropy of elastic constants, the dispersion of acoustic waves toward Brillouin zone boundaries, and optical frequencies that are much greater than the predicted Debye frequency. We also corrected for the volume dependence of the mode frequencies [Kieffer, 1982] and, at high temperatures, for the anharmonicity of the vibrational modes [Gillet et al., 1990]. As there is insufficient spectroscopic data to constrain the anharmonicity of the individual modes, we used a shock temperature measurement on fused quartz in the stishovite phase [Lyzenga et al., 1983; Boslough, 1988] to determine an average linear anharmonic correction at high temperatures. The overall effect of the anharmonic contribution is to increase the heat capacity of silica beyond the Dulong-Petit limit; for example, we find the heat capacity of stishovite at 70 GPa and 4500 K is 3.65(0.15)kB per atom. Details of the Kieffer model parameters and the anharmonic correction are presented in the auxiliary material. The entropy increase upon isochorically heating silica from the principal isentrope at TA = 334 K to the melting curve at TB = 4500 K is ΔSAB = 3158 ± 50 J kg−1 K−1.
 The Clausius-Clapeyron relation was used to solve for the entropy of melting stishovite at 70 GPa and 4500 K. The slope of the melting curve is 10.8 K GPa−1 at 70 GPa [Luo and Ahrens, 2004]. To determine the specific volume of liquid silica at 70 GPa and 4500 K, we compiled and fit the Hugoniot data for fused silica at pressures of 70 GPa to 1000 GPa [Marsh, 1980; Lyzenga et al., 1983; Brygoo et al., 2007],
where the covariance matrix is
The volume of liquid silica on the fused quartz Hugoniot at 70 GPa is VC = 2.153(48) × 10−4 m3 kg −1, consequently the relative volume increase upon melting stishovite at 70 GPa and 4500 K is given by (VC − VB)/VB= 1.3%. From the Clausius-Clapeyron relation, the entropy of melting is then ΔSBC = 253 J kg−1 K−1, which is within 15% of the empirical Rln(2) rule for the entropy of melting observed in many metals and silicates [Wallace, 2002; Luo and Ahrens, 2004]. As the uncertainties in the volumes are on the order of the volume change of melting, formal error propagation would lead to what we consider to be an unreasonably large error in the entropy of melting. The systematics of melting constitute a constraint on the uncertainty that is not captured by formal error propagation. For the purpose of further error propagation, we assign an illustrative uncertainty in the entropy of melting stishovite at 4500 K to be 150 J kg−1 K−1, which is very likely a conservative estimate given the previously mentioned systematics in the entropy of melting.
 Next, we determine the point where the fluid isentrope from the base of the fluid region on the fused quartz Hugoniot (state C) intersects the quartz Hugoniot using Equation (2). The dominant source of uncertainty in the intersecting point is in the Grüneisen parameter of fluid silica. There are a number of sound velocity measurements in the fluid phase; using the Hugoniots for fused quartz (Equation (4)) and quartz [Knudson and Desjarlais, 2009], we have reanalyzed and tabulated the corresponding Grüneisen parameters in Table S.4 (in Text S1) in the auxiliary material. As there is a great deal of scatter in the Grüneisen parameters over such a small range of volumes, we used the weighted average, γ = 0.7 ±0.3. With the experimental average of the Grüneisen parameter, we found the isentrope from state C intersects the quartz Hugoniot at TD = 4790 ± 140 K, which is very close to the melting temperature of 4800 K at 113 GPa [Luo et al., 2004]. Hence, this fluid isentrope intersects at our desired reference state (D), the base of the fluid region on the quartz Hugoniot.
 In summary, the entropy where the fused quartz Hugoniot leaves the melting curve, SC, is the sum of the initial entropy of stishovite, the entropy upon heating stishovite from the isentrope to the melting curve, ΔSAB, and the entropy of melting, ΔSBC:
 With our estimate of the Grüneisen parameter, the entropy at the base of fluid region on the quartz Hugoniot is the same as at the base of the fluid region on the fused quartz Hugoniot except with a slightly higher uncertainty: SD = 3820 ± 168 J kg−1 K−1. The entropy on the quartz Hugoniot in the fluid region, as presented in Figure 2, is the sum of reference state SD and the entropy change along the Hugoniot, determined using Equation (1) and data from Knudson and Desjarlais  and Hicks et al. . The intermediate states calculated to determine the absolute entropy of the fluid region on the quartz Hugoniot are given in Table 2. The absolute entropy as a function of pressure along the quartz Hugoniot from 113 to 800 GPa is given by
where P is the pressure along the Hugoniot in GPa, and SH,Qtz is the entropy along the quartz Hugoniot in J kg−1 K−1. The absolute uncertainty in the entropy along the quartz Hugoniot is given by
 The entropy on the fused quartz Hugoniot in the fluid region is the sum of reference state SC and the entropy change along the fused quartz Hugoniot, determined using Equation (1) and data from Lyzenga et al.  and Boslough . The entropy as a function of pressure along the fused quartz Hugoniot from 70 to 110 GPa is given by
For this relatively small pressure range, the uncertainty in the entropy along the fused quartz Hugoniot is approximately equal to the uncertainty in the entropy at the base of the fused quartz Hugoniot (160 J kg−1 K−1).
 Next, we present a calculation of the reference entropy states at ambient pressure.
2.2. Boiling Point Entropy of Silica
 Although slightly arbitrary for all impacts except on Earth, the entropies of the liquid and vapor phases at the 105 Pa boiling point are often used to calculate the criteria for incipient and complete vaporization upon release from shock. As there are no direct measurements of the boiling point of silica at 105 Pa, the value must be calculated using thermodynamic data that is extrapolated from much lower temperatures. Previous calculations of the boiling point of silica used outdated thermodynamic data for the liquid phase; for example, the widely cited JANAF thermochemical tables [Chase, 1998] uses heat capacity data for silica liquid from Wietzel .
 Here, we calculated the boiling point for silica following a similar method to that of Schick . In our analysis of chemical equilibrium, we included the reactions:
The energy cost for creating atomic silicon is too high for the partial pressure of silicon to effect the calculation of the boiling point, and the Si2O2 dimer is unstable at high temperatures [Schick, 1960]. We used enthalpy data for liquid silica from Richet et al.  and Tarasov et al. , which extend up to 2400 K. Thermodynamic data for the vapor species were obtained from the JANAF thermochemical tables [Chase, 1998].
 The partial pressures of the vapor species considered in the reactions described in Equations (10)–(12) sum to 105 Pa at T = 3177 ± 115 K. This boiling point is approximately 100 K higher than the analysis by Schick and only 20 K higher than the M-ANEOS result of 3157 K [Melosh, 2007] (shown in Figure 2). The physical reason for the higher boiling point is related to a higher average heat capacity and hence entropy in the liquid than previously thought, causing liquid silica to be more stable at higher temperatures. The temperature difference between this boiling point calculation and those calculations that used the JANAF thermochemical table [Chase, 1998] is not more pronounced is because the heat capacity data for liquid silica [Wietzel, 1921], cited in the JANAF table, is erroneously high. A discussion on the heat capacity of liquid silica is presented in the auxiliary material.
 The partial pressures and specific entropies for the vapor species are given in Table 3. At 3177 K, the entropy of the vapor is 7254 ± 8 J kg−1 K−1, and the entropy of SiO2 liquid is 3552 ± 70 J kg−1 K−1. Thus, the enthalpy of vaporization is 11770 ± 950 kJ kg−1. A detailed description of the boiling point calculation is presented in the auxiliary material.
Table 3. Composition of Silica Vapor at the Boiling Point: 3177 K and 105 Paa
Partial Pressure [105 Pa]
Specific Entropy [J kg−1 K−1]
Uncertainties in the specific entropies, taken from Schick , are estimates and are not experimentally determined.
6660 ± 9
8960 ± 8
13182 ± 12
6023 ± 33
 With the new reference values for the entropy along the Hugoniot and the entropies of the liquid and vapor in the released state, the shock states are more accurately linked to final release states at 105 Pa (Figure 2). However, there are insufficient thermodynamic data to calculate the liquid-vapor boundary, using the free energy minimization described above, at pressures other than 105Pa. Next, we present shock-and-release experiments to probe states along the liquid-vapor curve at pressures and temperatures up to the critical point.
3. Shock-and-Release Experiments
 To reach the phase space near the liquid-vapor curve of silica, samples of c-cutα-quartz (initial density 2.65 g cm−1) were shocked to a high-pressure fluid state. Once the shock wave reaches the downrange free surface, a zero pressure boundary condition, the shock wave is reflected as a centered rarefaction wave, which decompresses the fluid. The experiments were performed using the Janus laser at the Jupiter Laser Facility of Lawrence Livermore National Laboratory and utilized the line-imaging velocity interferometer systems for any reflector (VISAR [Barker and Hollenbach, 1972; Celliers et al., 1998]) and the streaked optical pyrometer (SOP [Spaulding et al., 2007]). Shock pressures and temperatures up to ∼300 GPa and ∼15,000 K were generated by direct laser ablation.
 The Janus laser is a two-beam Nd:glass laser capable of delivering 350 Joules per beam at the frequency doubled wavelength of 527 nm for pulse lengths of 2 to 10 ns. The laser beams were focused onto the target and a random phase plate was used to create an ∼700μm diameter spot. A randomizing phase plate maps any low-frequency spatial perturbations in the beam intensity into a uniform high-frequency speckle pattern. The intensity of the laser pulse was temporally shaped to generate a quasi-steady shock in the target. We refer the reader toSwift et al. for a general discussion of laser-driven shock compression.
 The ablation surface of each quartz sample was coated with 100 nm of aluminum to protect downrange optical components from laser light, which was only necessary prior to the silica sample becoming opaque due to plasma generation at the ablation surface. The dimensions of the target, 80 to 120 μm thick and 3 mm diameter, were chosen to maximize the area under uniaxial strain and the thickness of the spall layer that formed when the rarefaction wave from the downrange free surface interacted with the rarefaction wave created at the ablation surface. Hafnium oxide was used as an anti-reflective coating on the downrange free surface on most targets to decrease unwanted reflections in the VISAR diagnostic. However, post-shock temperature measurements were not made during experiments with anti-reflective coated samples so as to avoid any thermal contamination from the shocked coating material.
 For shock pressures above the melting curve, the shock wave was partially reflecting to the VISAR laser wavelength (532 nm) [Hicks et al., 2006] and can be used as the reflecting surface for interferometric velocimetry. Two line-imaging VISAR's, configured with different velocity sensitivites, were used in each experiment; the uncertainties in the absolute magnitude of the shock velocity measurement were approximately 0.35 and 1.14 km s−1, respectively. The VISAR signals were recorded on streak cameras with temporal resolution of ∼300 ps and total duration of 50 ns. An example line-VISAR image is shown inFigure 3. A timing fiducial was used to temporally synchronize the streak cameras.
 The known Hugoniot of quartz [Knudson and Desjarlais, 2009] was used to determine the shock pressure from the measured shock velocity (in km s−1):
For experiments where the reflectivity at the free surface was on the order of the reflectivity of the shock front, a superposition of fringes was detected on the line-VISAR system. One set of fringes originated from the reflection off the moving shock front (the signal) and the other from the reflection off the stationary interface between quartz and vacuum (called a ghost fringe). The method for subtracting the ghost fringes is described in supplemental material ofCelliers et al. . In our analysis, we used the reflectivity and temperature of shocked quartz as a function of shock velocity. The shock front reflectivity measurements from Hicks et al.  was revised in Celliers et al.  (supplemental materials) to
which is valid for shock velocities from 10 to 32 km s−1. The corresponding shock temperature (in K) TH,Qtz and temperature uncertainty δTH,Qtz along the quartz Hugoniot are given by
 As quartz is transparent to visible radiation, thermal radiation emitted from the shock front was recorded as a function of time and space, along a line across the target, using the streaked optical pyrometer. Due to the expected low emission intensity in the post-shock state, the streaked optical pyrometer was spectrally integrated from 450 to ∼800 nm. As in previous pyrometry experiments [Spaulding et al., 2007; Spaulding, 2010], calibration of the SOP was made relative to the known shock temperature and reflectivity in quartz (Equations (14) and (15)) [Hicks et al., 2006; Celliers et al., 2010]. The uncertainty in the calibration is approximately 11% over the range of temperatures considered here; a complete description of the calibration technique is presented in the auxiliary material.
 Upon breakout of the shock wave at the free surface, the radiating temperature of the sample should decrease almost instantaneously because of adiabatic cooling of the material. Transport of heat from the shocked material to the released material has been shown to keep released material from following a purely isentropic path in some experiments [Celliers and Ng, 1993]. However, as the expected ionization state in the decompressed silica is low, thermal conduction was likely insignificant for the length scale (and hence timescale) of interest, we assumed isentropic decompression.
 After shock breakout, the SOP signal intensity decays exponentially to a constant nonzero value with a time constant of ∼5 ns; example data are shown in Figure 4. The dominant source of the exponential, rather than stepwise, decay profile is due to convolution of the thermal emission in the shocked state by the camera-specific point spread function. To avoid contamination from the thermal emission in the shock state, we determined the post-shock thermal emission by averaging over a 10 ns period in the SOP intensity from at least 8 ns after shock breakout (the delay varied slightly depending on the intensity of the shock wave). The time period of 10 ns was chosen for consistency, and varying the period does not effect the measurement significantly as the post-shock thermal emission is approximately constant on these timescales. The reported uncertainties in the post-shock intensity were determined from both the spatial and slight temporal variation in thermal emission.
 In calculating an apparent temperature of the post-shock stateTa, we made the approximation that the decompressed material radiated as a blackbody. We expect this approximation to be valid based upon two lines of reasoning. First, spectrally resolved post-shock thermal emission measurements on silica shocked up to 130 GPa are well fit by a blackbody function [Boslough, 1988]. Similarly, in our experiments, we observed negligible reflected light from post-shock states with the VISAR diagnostic.
 Second, we calculated absorption coefficients in the visible range for SiO2 liquid and vapor using first principles molecular dynamics (FPMD) simulations based on density functional theory (DFT) using the Vienna Ab initio Simulation Package (VASP) [Blochl, 1994]. The results are shown in Figure 5, for a range of states around the critical point estimate from M-ANEOS of 5398 K and 0.549 g cm−3. The calculated absorption coefficients for densities of 0.55 g cm−3 and greater are weakly dependent on wavelength and suggest an optical depth of less than about 0.5 μm. If phase separation occurred on the timescale of the experiment and the low-density vapor expanded ahead of the decompressed liquid, then the optical depth of the leading vapor would be negligible until a layer greater than about 200μm formed. Below (section 3.2), we discuss the results from a series of stagnation experiments that do not show significant fast-moving vapor ahead of the liquid. Thus, we infer that the vapor did not completely separate on the timescale of the experiments, and consequently, the decompressing vapor did not shield emission from the liquid-vapor interface. To first order, the problem can be simplified to emission from an opaque interface, where radiation is emitted from the interface between liquid and vapor. A more detailed analysis of the radiative transfer problem is presented insection 5.4.
 Post-shock SOP counts and calibrated apparent post-shock temperatures are presented inTable 4. Post-shock temperatures are associated with shock states based upon the shock velocity immediately prior to shock breakout at the free surface. For the lowest pressure experiment (e100728s4), the intensity of the free surface reflection was significantly greater than the shock wave VISAR signal, preventing a simultaneous shock velocity measurement. In this case, the shock state was determined from the shock temperature prior to breakout using the calibration developed from the higher-pressure experiments. Because the signal was expected to be low for this experiment, the gain on the SOP was increased from 30 to 40 and the calibration was adjusted accordingly.
Table 4. Apparent Post-shock Temperature of Silica From Laser and Gas-Gun Shock-and-Release Experimentsa
 Below, in section 5, we discuss the kinetics of phase separation and the radiative transfer through the decompressing silica in more detail.
3.2. Stagnation Experiments and Post-shock Density
 To estimate the average density of the decompressing material, we performed a series of stagnation measurements on quartz shocked to pressures of about 200 and 340 GPa. Upon breakout of the shock wave at the free surface, the material uniaxially expanded across a gap of known distance and stagnated against an aluminized LiF window. The velocity at the interface between the stagnating silica and the LiF window was measured using the line-VISARs. These experiments were qualitatively similar to the gas-gun experiments byAsay and Trucano , Chhabildas et al. , and Alexander et al. . The stagnation experiments complement the post-shock temperature experiments by probing the density gradient in the expanding material, which is important for interpreting the source of the thermal emission (seesection 5.4).
 For each target, a spacer of known thickness (approximately 50, 100, and 200 μm) separated the downrange free surface of the quartz from a 1-mm thick LiF window. A schematic of the target design is presented inFigure 6. A 5-μm layer of aluminum coated the uprange LiF surface, forming both a reflecting interface for the VISARs and a thermal barrier as the transparency of LiF is strongly temperature dependent. Based upon two-dimensional hydrodynamic simulations using the CTH code [McGlaun et al., 1990], for these gap distances, the released silica maintains conditions of uniaxial strain over the duration of the stagnation measurement.
 Because the aluminized LiF prevents direct observation of the shock in the quartz sample during the stagnation experiments, the shock pressures were determined by a separate series of experiments where the shock velocity was directly observed for varying laser pulse intensities. In combination with radiation hydrodynamics simulations using the Hyades code [Larsen and Lane, 2002], we derived a scaling law for the ablation pressure as a function of laser intensity for direct ablation of quartz with a 100-nm aluminum coating:Ps = 5IL0.78, where the ablation pressure Ps is in GPa and the laser intensity IL is in PW m−2. The shock pressure of each stagnation experiment was derived from the measured laser intensities; each set of experiments should differ by less than 10%, which is similar to the magnitude of variation in the quasi-steady shock pressure during a single experiment.
 The average shock velocities experienced by the silica spall layer were 12.1 ± (0.35, 0.45) and 15.3 ± (0.20, 0.45) km s−1, respectively. The corresponding shock pressures are 199 ± (14, 17) and 338 ± (10, 24) GPa, where the first uncertainty is the variation about the mean and the second term is the random uncertainty. Figure 7 presents the time at which the shock breaks out of the quartz sample and the time at which the decompressing silica impacts the LiF window for different gap thicknesses.
 The mean impact velocity Vimp of the silica onto the LiF window was determined from a linear fit to the transit time versus gap distance. The time of shock breakout was determined by the midpoint in the loss of reflectivity, whereas the time the stagnating silica impacts the LiF window was taken from the midpoint in the rise of the interface velocity corrected for the transit time through the aluminum coating. As there appears to be a systematic offset in the different time measurements, the time of shock breakout was not included in the linear fit.
 Example stagnation velocity profiles at the silica-LiF interface are presented inFigure 8. The VISAR fringe constant for the LiF window is 8.311 km s−1, and the random uncertainty in the fringe position results in a particle velocity uncertainty of approximately 0.4 km s−1. With increasing gap thickness, the risetime of the interface velocity increases because the density gradient decreases with expansion distance. The observed nanosecond precursor signal could represent vapor a few tens of microns ahead of the main density gradient, which would not be sufficient to absorb significant radiation from the liquid-vapor mixture.
 To obtain the average density in the expanding silica from the interface velocity measurements, we analyzed the stagnation data in a manner similar to a reverse impact experiment. By assuming the density of the spalled silica layer is constant, the density of the spall layer just prior to impact can be determined using the impedance matching method,
where ρ0,F is the average density in the decompressing fluid immediately prior to impact, PLiF is the shock pressure induced in the LiF window, uP,SFis the particle velocity in the re-shocked silica fluid, andUS,SFis the shock velocity in the re-shocked silica fluid. The shock pressure in the LiF was determined from the measured particle velocity and the known LiF Hugoniot [Ahrens and Johnson, 1995a]. The particle velocity in the re-shocked silica fluid was equal to the impact velocityVimp less the measured interface velocity uint.
 The shock velocity in the re-shocked silica fluid was not observed. As the Hugoniot of silicate fluids starting near the critical point is not known, we needed to make a prediction about theUS,SF − uP,SFsystematics of the decompressed silica fluid. There was not enough information to estimate a Hugoniot for these decompressed fluid states based on thermodynamic measurements; consequently, we used model Hugoniots from the M-ANEOS equation of state for silica to estimate a bound on the shock velocity for a given particle velocity. For initial densities ranging from 0.007 to 1.3 g cm−3 and temperatures from 4000 to 6000 K, the shock velocity varies by only about 20%.
 Using the M-ANEOS model Hugoniots, we fit the bulk sound speed,C0,SF, and slope of the US,SF − uP,SF relation, sSF, with a linear function of the initial density [c.f., Anderson, 1997]:
where density is in g cm−3 and C0,SF is in km s−1. We assigned an uncertainty in the shock velocity of 20%, which represents the envelope of model shock velocities, for a given particle velocity, for a fluid parcel of initially 25% to 75% vapor by mass.
 Under the impedance matching assumption (Equation (17)), the US,SF − uP,SF relation presented in Equations (18) and (19), and the LiF Hugoniot [Ahrens and Johnson, 1995a], we solved for the average density of the silica fluid immediately prior to impact, ρ0,F. The results are presented in Table 5. As expected, for higher initial entropies, the released state had a lower density. Note that significant vaporization was occurring on the timescale of a few nanoseconds as the average density decreased by a factor of 3 to 4.
Table 5. Average Post-shock Densities of Silica From Shock-and-Release Stagnation Experiments
Vimp [km s−1]
uP,LIF [km s−1]
ρ0,F [g cm−3]
Release From 199 ± 22 GPa
13.8 ± 0.5
4.6 ± 0.6
137 ± 16
1.16 ± 0.29
13.8 ± 0.5
4.0 ± 0.5
111 ± 12
0.86 ± 0.21
13.8 ± 0.5
2.2 ± 0.45
47 ± 8
0.28 ± 0.08
Release From 338 ± 30 GPa
18.0 ± 0.7
5.5 ± 0.5
182 ± 15
0.89 ± 0.20
18.0 ± 0.7
4.0 ± 0.55
111 ± 13
0.45 ± 0.11
18.0 ± 0.7
3.25 ± 0.45
136 ± 8
0.30 ± 0.07
 The shock pressure in the LiF was determined assuming the measured particle velocity represents a Hugoniot state. Although this assumption is not strictly true for the experiments that show a ramp loading profile, the Hugoniot approximates an isentrope in pressure-volume and pressure-particle velocity space, within 7% in pressure, at the moderate pressures of the stagnation experiments. Also, there could be a systematic uncertainty inρ0,F related to the US,SF − uP,SFrelation derived from M-ANEOS. However, the conclusion that significant vaporization occurred on the timescale of the experiment is robust.
 More sensitive stagnation experiments are necessary to accurately probe the density gradient of a vaporizing fluid. Next, we compare the experimental results to the model liquid-vapor curve for silica.
4. Revised Liquid-Vapor Curve of Silica and Post-shock Temperature Data
 As the current M-ANEOS model liquid-vapor curve for silica does not lie within error bars of the boiling point entropy calculation from this study (Figure 2), we modified the M-ANEOS model parameters in the expanded region to fit the pressure, temperature, and entropy at the calculated boiling point (section 2.2) and measured triple point (1920 ± 50 K and ∼2 Pa) [Mysen and Kushiro, 1988].
 The model liquid-vapor curve required only slight modifications to the M-ANEOS parameters in the expanded region to fit both the boiling point and triple point (Evap = 1.228 ± 1011 erg g−1, a_exp = 1.6, Tdebye = 660 K, Grun = 0.77, C53 = 0 dynes cm−2, C54 = 0). Note that these parameters should not be used in M-ANEOS to model the equation of state surface outside of the liquid-vapor dome. The model liquid-vapor curve is tabulated in theauxiliary material (Table S.5 in Text S1). The critical point from our model occurs at TCP = 5130 K, ρCP = 0.508 g cm−3, PCP = 0.13 GPa, and SCP = 5150 J kg−1 K−1.
 In Figure 9, we present the new post-shock temperature data for silica (Table 4) in temperature-entropy space, assuming isentropic decompression from the Hugoniot state. Post-shock temperature measurements on quartz and fused silica from gas-gun experiments which achieved fluid shock states are also shown [Boslough, 1988; Lyzenga et al., 1983]. The post-shock temperatures follow the new model liquid-vapor curve temperature up to states near the model critical point, where they begin to diverge slightly below the equilibrium liquid-vapor curve.
Boslough considered the possibility of the release isentropes from the shock states intersecting the liquid-vapor dome. However, without the stagnation measurements to show vaporization was occurring on the timescale of the experiment,Boslough  assumed the fluid would supercool below the liquid–vapor phase boundary, as suggested in Zel'dovich and Raizer . Consequently, Boslough used the post-shock temperatures to estimate post-shock densities and the fluid Grüneisen parameter. However, the form of the Grüneisen parameter assumed byBoslough , which is usually correct for solids, is incorrect for fluids [e.g., Stixrude and Karki, 2005] and led to a significant overestimate of the post-shock densities (∼3.76 g cm−3).
 Given the entropy in the shock states of Boslough and our model liquid-vapor curve, the post-shock density of the liquid should have been ∼2.00 g cm−3, which is similar to the density of the liquid at the boiling point (∼2.05 g cm−3 [Hudon et al., 2002]). For significantly higher entropies, such as in our stagnation experiments discussed in the previous section, the post-shock density would have been lower and would decrease rapidly with time due to a rapidly increasing volume fraction of expanding vapor. For example, for release from 200 GPa, the density of the liquid where the release isentrope intersects the liquid-vapor dome is 0.72 g cm−3, which is comparable to the average density determined in the stagnation experiments at a gap distance of 100 microns, 0.86 g cm−3, and significantly greater than the average density determined at a gap distance of 200 microns, 0.28 g cm−3. Detailed comparisons between the average density determined in the stagnation experiments and the density of the liquid-vapor mixture were impractical due to the complicated experimental geometry; however the conclusion that bulk vaporization was occurring on the timescale of the experiment is robust.
 Although the stagnation experiments were an integrated measurement of the momentum transfer between the silica and the LiF, post-shock temperature measurements were a significantly more local measurement. In the next section, we investigate the process of emission from a vaporizing fluid and assess how the apparent post-shock temperature should compare to the liquid-vapor curve.
5. Analysis of an Isentropically-Expanding Liquid-Vapor Mixture
 Based on the relative absorbance of silica liquid and vapor, in section 3.1we argued that the post-shock thermal emission originated at the liquid-vapor interface. Here, we show that although the opacities of the liquid and vapor are uncertain, the dominant emitter was the silicate liquid and not the rapidly cooling low density vapor plume. To make this argument, we first estimate the dominant mechanism for phase decomposition. Then, we present the density and temperature profiles in the releasing fluid as a function of position. Finally, using the release profiles and estimates of the absorption coefficients, we performed a radiative transfer calculation to determine the emitting region in the released material.
5.1. Phase Decomposition
 Phase decomposition is likely to occur by fragmentation, nucleation and growth, or spinodal decomposition. The dominant mechanism will depend on where the fluid release isentrope intersects the liquid-vapor dome and the relative timescales for each process [Povarnitsyn et al., 2008], where the fastest mechanism wins. Here, we explore the feasibility of each phase decomposition mechanism on the timescale observed in the stagnation experiments. Understanding the mechanisms for vaporization will aid in interpretation of the post-shock temperature measurements (section 5.4) as well as the design of future shock-and-release experiments.
 For a release isentrope that enters the liquid-vapor dome on the liquid side of the critical point (Figure 2), significant undercooling could allow the material to go into tension. If the tensile stress exceeds the spall strength of the liquid for sufficient time, the material will undergo fragmentation, also called cavitation. Grady  derived a timescale for a liquid to undergo fragmentation if the spall strength has been exceeded,
where ς is the surface tension, ρ is the density of the liquid, is the strain rate, and cs is the adiabatic sound speed. The surface tension decreases to zero at the critical point temperature TCP in the form [Povarnitsyn et al., 2008]
The coefficient ς0 is calibrated using the surface tension of silica liquid at ∼2000 K, ς = 0.3 N m−1 [Boca et al., 2003].
 In the shock-and-release experiments, the decompressing strain rate changed dramatically with distance from the free surface. For material within 10μm of the free surface, the decompression strain rate can be approximated to an order of magnitude by assuming a constant decompression rate from the shocked density ∼6 g cm−3 to a metastable liquid density ∼1.5 g cm−3 in a timescale of a few nanoseconds, which leads to a strain rate of ∼4 × 108 s−1. Assuming a sound speed of 1 km s−1 (consistent with the Hugoniot bulk sound speed), we found the time scale for cavitation to be 10 to 100 picoseconds for a range of temperatures below the critical point.
 If the isentrope intersects the liquid-vapor dome near the critical point (Figure 2), the metastable liquid isentrope must undergo significant supercooling before it reaches a state of negative pressure. One can use the theory of homogeneous nucleation to model the nucleation of bubble droplets in the adiabatic decompression of fluids into the liquid-vapor dome [Povarnitsyn et al., 2008]. The theory can also be used to predict the lag time, τnucl, between the fluid parcel becoming metastable and when nucleation and growth begins [Balluffi et al., 2005; Povarnitsyn et al., 2008], which is the most important factor in predicting the dominant mechanism for phase decomposition. The lag time is given by
where Λ ≈ 1010 s−1 is the kinetic coefficient [Tkachenko et al., 2004], n is the number density of particles, V* is the volume under consideration, W = 16πς3/ΔP2 is the work required to form a critical size bubble, and ΔP is the difference between the pressure in the metastable state and the pressure on the liquid–vapor curve at the same temperature. Given the strain rates in these experiments, a reasonable volume element to consider is (10 μm)3. The value for ΔP is dependent upon the timescale, but for the purposes of estimating the timescale for nucleation, we assumed ΔP = 0.01 GPa and T = 5000 K, which resulted in τnucl ≈ 10 ps, a timescale that would suggest nucleation of vapor droplets occurred well within the time frame of the experiments. This timescale is strongly dependent on the surface tension and ΔP. Consequently, the uncertainty in τnucl was poorly constrained as the surface tension is extrapolated well beyond the conditions of any experimental data and ΔPwould depend strongly on the true release isentrope and the liquid-vapor curve.
 Spinodal decomposition occurs when the free energy surface has negative curvature with respect to the extrinsic variable of interest (e.g., chemical composition or density). The early stage growth by spinodal decomposition in two-component chemical systems is relatively straightforward to treat analytically as one need only consider the equation of motion for the diffusion of atoms and mass conservation. However, to predict the dynamics of spinodal decomposition in density space, one need also consider the hydrodynamic equations of motion, which do not lend themselves to simple analytical solutions. Although few numerical studies have considered spinodal decomposition upon quenching into the liquid-vapor dome, recent two- and three-dimensional simulations for quenches into the liquid-vapor dome of a van der Waal's fluid [Lamorgese and Mauri, 2009] provide useful constraints on the timescales for decomposition.
Lamorgese and Mauri  find that spinodal decomposition process occurs in three stages. First, upon instantaneous quenching there is a time delay where no detectable phase separation takes place as the thermal density perturbations are not strong enough to overcome the surface tension in the fluid. The second stage is defined by a rapid phase separation to a state with liquid and vapor phases near the respective equilibrium densities. The third stage represents an asymptotic approach toward equilibrium where coarsening also takes place.
 Most important to predicting whether phase separation occurs by nucleation and growth or spinodal decomposition is the time delay between quench and rapid phase separation. For the van der Waal's fluid, Lamorgese and Mauri  find that the time delay depends extremely weakly on the surface tension, a useful result as the surface tension is poorly constrained for silicate fluids. As noted in Vladimirova et al. , the time delay depends predominantly on the depth of the temperature quench with the intensity of thermal noise having a minimal effect on the time delay. Lamorgese and Mauri  find that for a quench of 10% below the critical point temperature and a range of unstable densities, the time delay is on the order of 1 to 10 scaled time units. In their simulations, time is scaled by a macroscopic length scale divided by the sound speed in the medium. Consequently for a system size of ∼10 μm with a sound velocity of 0.1 to 1 μm ns−1, the time delay before phase separation occurs in a 10% quench will be 10 to 1000 ns, significantly longer than the timescales expected for homogeneous nucleation at much smaller quenches.
 Based on the relatively sparse data summarized above, we suggest that phase separation occurred by nucleation and growth rather than spinodal decomposition for the shock and release experiments presented here. However, future experiments are needed to determine the true kinetics and particle size distribution upon vaporization.
5.2. Post-shock Temperature Buffering
 The decrease in temperature with density during isentropic decompression from the shock state can be determined if the Grüneisen parameter of the fluid is known as a function of volume. However, upon the isentrope entering the liquid-vapor dome, it is no longer appropriate to integrate using the simple functional form ofEquation (2)as the isentropic path is dominated by the relative mass of liquid and vapor. In effect, for an isentropically expanding mixture the temperature is buffered upon entering the liquid-vapor co-existence region. The equilibrium density-temperature path through the liquid-vapor dome can be determined if the entropy and specific volume of the liquid and vapor (Sliq, Svap, Vliq, and Vvap) are known as a function of temperature. Upon entering the liquid-vapor dome, the entropy and volume of the mixture,Sm and Vavg, are given by
where χvapis the mass fraction of vapor. Using the Clausius-Clapeyron relation,Equations (23) and (24) can be combined into a more useful form,
where is the slope of the liquid–vapor curve at temperature T.
 Near the critical point, Vliq changes dramatically as a function of temperature; however, below about 0.95TCPthe volume of the liquid is relatively constant and negligible with respect to the volume of the vapor at the same temperature. For release isentropes that intersect the liquid-vapor curve below about 0.95TCP, the volume change in the liquid is small and consequently the entropy change in the liquid can be approximated assuming a temperature-independent isochoric heat capacityCV,
where TLV,mis the temperature where the isentrope entered the liquid-vapor dome. As most materials follow an Arrhenius law for the saturated vapor pressure as a function of temperature, the boiling point and triple point constrain the slope of the liquid-vapor curve,
where A1 and A2 are constants, 5.58 × 108 Pa K and 5.85 × 104K, respectively, for the liquid-vapor curve of silica fromMelosh .
 Consequently, the average specific volume for the isentrope within the co-existence region has the simple functional form,
This analytic model is compared to M-ANEOS model isentropes inFigure 10. For isentropes that intersect the liquid-vapor curve near the critical point, the approximate solution inEquation (28)breaks down as the entropy change along the liquid side of the vapor curve is no longer dominated by the change in temperature but also by the significant decrease in volume upon cooling. However, even near the critical point, once the isentrope enters the liquid-vapor dome, the temperature decrease is only ∼10% for over a 10-fold increase in volume.
 The significant temperature buffering is caused by the statistical need to minimize the free energy and thereby make vapor. For the post-shock temperature experiments described insection 3.1, even though the leading edge of the fluid was decompressing rapidly, the temperature gradient in the vaporizing liquid-vapor mixture was small.
5.3. Post-shock Density Profiles
 To be able to calculate the radiative transfer through the expanding liquid-vapor mixture, in addition to the temperature profile, we need the density profile and information about the mixture geometry. Upon breakout of a steady one-dimensional shock at the free surface, a centered rarefaction wave propagates back into the silica sample. The compressibility of a liquid-vapor mixture is significantly lower than the pure liquid phase as most of the strain is taken up by the more compressible vapor phase. The discontinuity in compressibility as the material releases into the liquid-vapor dome results in a change in the slope of an isentrope in pressure-density space. This compressibility discontinuity has significant implications for the density profile of a material undergoing isentropic decompression.
 As an example, we calculated the rarefaction fan from a 200-GPa shock breaking out at a free surface using the method of characteristics [e.g.,Zel'dovich and Raizer, 1966]. The M-ANEOS model Hugoniot and release isentrope from 200-GPa are presented in pressure-density space inFigure 11a. The slope of a characteristic is given by
where h is the Lagrangian coordinate, ρi is the material density along characteristic i, cs,i is the Eulerian sound speed at density ρi, and ρ0 is the density in the Lagrangian frame of reference. In Figure 11b, each characteristic is separated in density by 0.1 g cm−3. The significant and abrupt decrease in sound velocity upon the isentrope entering the liquid-vapor dome leads to a separation of rarefaction waves. As a result, a plateau forms at the density where the isentrope intersects the liquid-vapor dome and increases in thickness with time.
 We transformed the density profile in Lagrangian coordinates to Eulerian coordinates by considering conservation of mass,
where x0 is a constant of integration that will depend on the momentum transfer through the entire release path. The dependence of ρ on h was obtained from Equation (29). Equation (30)is valid for regions along the release isentrope where the Lagrangian sound speed increases monotonically with density. In regions where the Lagrangian sound speed decreases with density, a rarefaction shock wave will form, requiring the solution of the coupled conservation equations to predict the density profile. We then calculated the Eulerian density profile up to the stishovite transition, where the sound speed decreases with density on the M-ANEOS release isentrope, which is an artificial transition within the M-ANEOS model as it does not include a melting curve.
 In Figure 12, the results of our analytic Eulerian density profiles are compared with density profiles obtained from adiabatic release calculations using the shock physics code CTH [McGlaun et al., 1990] and the M-ANEOS silica EOS. The density profiles obtained from integratingEquation (30) were offset in space to match the CTH profiles as we did not integrate over the characteristics beyond the stishovite transition. The general shape of the hydrocode and analytic calculations are in excellent agreement.
 For isentropes that enter the silica liquid-vapor dome on the liquid side, the thickening plateau of material at the density of the phase boundary forms an optically-thick layer in the expanding mixture. Because the phase-boundary plateau is optically thick (Figure 5), thermal radiation from material at pressures above the liquid-vapor curve will not be observed after a period of about 200 picoseconds. Consequently, only temperatures at or below the temperature where the isentrope enters on the liquid side of the liquid-vapor curve were observed in thermal emission after shock-and-release.
 In contrast, for release isentropes that enter on the vapor side of the critical point, the phase-boundary plateau will be less pronounced and will eventually become nonexistent with increasing entropy. As a result, the observed thermal emission in such high pressure shock-and-release experiments would sample a larger range of temperatures than those at and below the liquid-vapor dome, and the interpretation of apparent temperatures would require a full radiative transfer calculation.
5.4. Radiative Transfer Model
 We developed a simple radiative transfer model to aid interpretation of the post-shock temperature data. Due to the large uncertainties in the droplet size distribution, concentration, and optical scattering properties within the liquid-vapor mixture, the model does not include any particle scattering. We considered a more conservative, yet simple, assumption that any photon that interacts with a liquid droplet is absorbed.
 Because of the high opacity of silica liquid, a high volume fraction of liquid droplets will be essentially opaque to any photons from uprange material. Consequently, even in experiments where a density plateau forms in the expanding mixture, the lower-density material ahead of the plateau may obscure thermal emission from material in the plateau. However, the difference between the temperature of the plateau and the temperature of a layer ahead of the plateau will be small due to the buffering by the phase boundary (section 5.2).
 To estimate the difference between the experimental apparent temperatures presented in Table 4and the true temperature along the liquid–vapor boundary, we considered a radiative transfer model that conservatively assumes liquid droplets are opaque and constrained the absorption coefficient of the vapor phase from the FPMD-DFT simulations presented inFigure 5. The probability of a photon being absorbed within a slice of the two-phase liquid-vapor mixture of thicknessdx is the sum of the probability of the photon interacting with a liquid droplet and being absorbed within the vapor phase,
where α(x) is the average absorption coefficient at position x, LMFP is the mean free path for a photon interacting with a liquid droplet, and αvap is the absorption coefficient of the vapor phase. The mean free path for a particle interacting with liquid spheres of diameter D0 and volume fraction Vliq/Vavg is given by
 The depth of emitting layers was found by calculating the thickness where the optical depth is unity,
xfs is the position of the vacuum interface and x(Ta) is the position of the emitting region within the liquid-vapor mixture. To calculatex(Ta) for a given entropy, we must know LMFP and αvapas a function of position, which requires a model for the absorption coefficient as a function of density and temperature and the pressure, density, and temperature states along the entire isentrope within the liquid-vapor mixture.
 Since this calculation was restricted to paths in the liquid-vapor co-existence region, we used our revised model liquid-vapor curve (section 4) to determine the thermodynamic properties as a function of position and time within the expanding silica, following the same procedure as in section 5.3. In Figure 13, we present the average density, the volume fraction of liquid, and the temperature as a function of position along a 4890 J kg−1 K−1release isentrope (generated by a 200 GPa shock) at 10 ns after shock breakout. We considered the profile at 10 ns after shock breakout to be consistent with the time period observed for the experimental post-shock temperature data.
 The 10-ns expansion profile was used to calculatex(Ta), using Equation (33)and the density and temperature-dependent absorption coefficients of silica vapor. We used our FPMD-DFT simulations to constrain the absorption coefficient over a wide range of densities and temperatures by modeling the frequency-dependent opacity with the semiconductor Drude model (seeauxiliary material). A radiometric measurement would record an apparent post-shock temperature,Ta, corresponding to the temperature at x(Ta), the location in the expanding mixture where the optical depth reaches unity.
 In Figure 14, we present the predictions for the apparent post-shock temperature as a function of entropy in free-surface shock-and-release experiments on quartz. With increasing entropy, post-shock temperature measurements initially follow the liquid–vapor curve until reaching an entropy close to critical point, where the observed post-shock temperatures will start to diverge below the liquid-vapor curve. The divergence occurs when the materials leading the phase-boundary plateau become optically thick.
 At an entropy state significantly greater than the critical point, the post-shock thermal emission will originate from materials above the liquid-vapor curve [Celliers and Ng, 1993]. In this case, the sum of materials at densities on and below the liquid-vapor dome is not optically thick. The point at which apparent temperatures would rise above the liquid-vapor dome depends strongly on the opacity of the fluid. Our model construction does not include the emission from states above the liquid-vapor curve; however, our model does calculate the point where the apparent temperature transitions from below to above the liquid-vapor dome.
6.1. Shock-and-Release to the Liquid-Vapor Phase Boundary
 Based on our analysis of an isentropically expanding liquid-vapor mixture, we find that post-shock temperature measurements are an excellent method for determining the temperature along the liquid–vapor curve at entropies below the critical point. The technique has increasing uncertainties as the entropy nears that of the critical point and above.
 Even with the simplifications made in analyzing the structure of isentropically-expanding silica, the model for apparent temperature observations after shock-and-release (Figure 14) closely follows the data obtained in the experiments (Figure 9). At entropies lower than the critical point, the gas-gun and laser-driven shock data are on or near the model liquid-vapor curve. In these cases, the temperature in the released silica was buffered by the phase boundary and absorption by lower-density material was negligible. As the shock entropies increase toward the critical point entropy, the experimental apparent post-shock temperatures flatten in a manner similar to the model predictions. There was some absorption by material with densities below the phase boundary, but the temperature of this material was only slightly below the phase boundary. The experimental post-shock temperature data also appear to intersect the liquid-vapor curve on the vapor side of the critical point at a similar entropy as predicted by the radiative transfer model (about 5500 J kg−1 K−1). Higher entropy shock-and-release experiments were not possible with the Janus laser. Our model suggests that such experiments would have recorded apparent post-shock temperatures above the liquid-vapor boundary.
 The bulk density measurements in the stagnation experiments provide confirmation of release into a liquid-vapor mixture: the densities were well below the expected metastable liquid density and the risetime in the LiF window increased with time. However, the limitations in the experimental configuration did not allow for determination of a detailed density profile in the decompressing material. Future stagnation experiments could be used to measure the density where the release isentrope enters the liquid-vapor dome and to estimate the kinetics of vaporization and condensation.
 Because of the abundance of thermodynamic data available for silica, we have been able to calculate a new liquid-vapor boundary and develop a model for post-shock temperatures independently of the data from our shock-and-release experiments. The model and data are self-consistent, which reinforces our confidence in both the physical model and the experimental technique.
 Given the uncertainties in the experimental data and our radiative transfer model, we do not consider the data to be an indirect constraint on the mean droplet size in the decompressing silica. The mean droplet size is expected to change dramatically as a function of entropy within the liquid-vapor dome [Tkachenko et al., 2004] and consequently future experiments that directly measure the droplet size and the density structure in the expanding material would increase the accuracy of the radiative transfer model and allow for more detailed interpretation of apparent post-shock temperature measurements.
 For the many materials where little data exists in phase space near the liquid-vapor boundary, shock-and-release experiments provide a means to obtain robust temperature measurements. The precision of post-shock temperature measurements can be improved in future work by including spectral resolution. Wavelength-dependent pyrometry is needed to be able to account for non-gray body emission. Higher temporal resolution would improve the radiative transfer calculations and allow for study of the absorption by the leading vapor. Future experimental configurations that resolve the density structure in the expanding material may also provide robust density measurements by identifying the predicted phase-boundary plateau.
6.2. The Criteria for Shock-Induced Vaporization of Quartz
 With our experimentally-validated model for the liquid-vapor curve of silica and new calculation of entropy on the Hugoniot, we determined the critical shock pressures required for incipient and complete vaporization upon decompression. Along the liquid-vapor curve, the temperature rises with ambient pressure; simultaneously, the entropies of the liquid and vapor converge with increasing ambient pressure (Figure 2). Therefore, the shock entropy required to initiate vaporization on release rises with ambient pressure while the shock entropy required for complete vaporization decreases.
 For the example shown in Figure 2, a 300-GPa shock reaches an entropy of 5697 ± 211 J kg−1 K−1. Upon decompression to 105 Pa, by the lever rule, 58 ± 6% of the silica is vaporized. In Table 6, we present the critical shock pressures, with 1-σ uncertainties, for incipient, 50%, and complete vaporization upon decompression to 105Pa and the triple point (∼2 Pa). For bodies with ambient pressures below the triple point, the decompressing silica will pass through the triple point and be buffered by the kinetics of freezing. Hence, the triple point serves as a useful reference point for shock-induced vaporization on airless bodies. The critical shock pressure for vaporization upon release to other ambient pressures may be determined by the entropy on the Hugoniot (Equation (7)) and the entropies on the liquid-vapor curve, which are tabulated in theauxiliary material (Table S.5 in Text S1).
Table 6. Criteria for Shock-Induced Vaporization of 298 Kα-quartza
SIV [J kg−1 K−1]
SCV [J kg−1 K−1]
Shock pressures and entropies for incipient (IV), 50%, and complete vaporization (CV) upon decompression to the triple point (∼2 Pa) and 105 Pa ambient pressure.
 The critical shock pressure for complete vaporization of silica is about a factor of two lower than the M-ANEOS model prediction inMelosh  and is approximately thirty percent less than a recent model by Kurosawa et al. that does not include the entropy of melting. As the M-ANEOS model liquid–vapor curve was near our revised model, the differences in critical shock pressures primarily reflect the lower entropy along the M-ANEOS model Hugoniot (Figure 2). The offset in entropy arises from a combination of factors, including the absence of a melting transition and the incorrect heat capacity in the fluid.
 The heat capacity reflects the manner of dissipation of the internal energy gained from the shock wave. Rather than shock energy being dissipated as thermal energy, and ionization at the highest energies, more of the shock energy in silica is being dissipated as disorder. The source of the disorder in the supercritical fluid is not perfectly clear; Hicks et al.  associate the higher heat capacity with dissociation and changes within the short range order in the molecular fluid. The contribution to the heat capacity from electrons is also likely greater than a thermally activated ionization model would predict as the concentration of free electrons increases dramatically upon melting, compression, and with increasing dissociation [Laudernet et al., 2004; Hicks et al., 2006]. As the physical mechanism for the high heat capacity is likely to be applicable to all silicate liquids that start out as a bonded fluid, we expect the critical shock pressures for vaporizing all silicate liquids to decrease significantly from previous estimates that assumed the Dulong-Petit limit for the heat capacity of the fluid.
6.3. Implications for Planetary Impact Events
 The pressure range under consideration, hundreds of GPa, is of particular importance to the end stage of terrestrial planet formation and subsequent impact cratering events. Peak shock pressures of about 100 GPa are achieved during impacts at velocities greater than ∼8 km s−1, comparable to the escape velocity of Earth-mass planets. Typical present-day impact velocities in the inner solar system range from about 5 to 40 km s−1 [Le Feuvre and Wieczorek, 2011]. Silica is a major component in planetary crusts and an end-member for the compositional range in planetary mantles. Our work finds that vaporization of silica is an important process during planetary impact events.
 A detailed analysis of the amount of vapor produced during planetary impacts is beyond the scope of this study. However, we estimated the magnitude of the effect of revising the critical shock pressure for vaporization. The pressure of a strong shock wave decays as one over the distance cubed from the impact point [Taylor, 1950; Croft, 1982]. The volume of material shocked to or greater than a given pressure scales by distance to the same power. Consequently, our revision of the critical pressure for vaporization (lower by a factor of two) yields approximately a factor of two greater vapor production.
 In the interior of planets, the temperatures and pressures are typically higher than the initial state of quartz studied here. For a higher initial temperature, as a first approximation, the entropy of heating may be added to the entropy on the 298 K Hugoniot. Second order differences in entropy, related to the different shock state attained when starting at a higher initial temperature and pressure, require a full equation of state model to calculate. A new multiphase equation of state model of silica, that may be used in future hydrocode calculations of impact events, is currently under development [Kraus et al., 2012].
 Given our improved understanding of the physics of shock-induced vaporization of silicates, it is very likely that previous studies have underestimated the amount of vapor produced during planetary impact events. Future simulations of planetary collisions using revised equation of state models that incorporate experimental constraints on the liquid-vapor coexistence region will more accurately reproduce the thermodynamics of planetary impact events.
 For many materials, the liquid-vapor phase boundary is inaccessible via static experimental methods. In these cases, shock-and-release experiments are a robust technique to determine states on the phase boundary. In this study, we investigated shock-induced vaporization of silica experimentally and theoretically. We find that shock-and-release experiments provide an accurate measurement of the temperature on the phase boundary for entropies below the critical point, with increasing uncertainties near and above the critical point entropy. For entropies below the critical point, we predict the development of a plateau in the profile of the decompressing material at the liquid density.
 In order to develop a theoretical model for the isentropic decompression of silica, we calculated reference entropy states using the most recent thermodynamic data. By performing an empirically constrained thermodynamic integration, we determined the absolute entropy at the point where the quartz and fused silica Hugoniots cross the melting curve. The entropy on the Hugoniots were determined using measured shock temperatures, pressures, and densities. The entropy on the quartz Hugoniot is significantly higher than the M-ANEOS model Hugoniot primarily because of the high heat capacity in the fluid. Using the M-ANEOS model construction for the expanded states, the liquid-vapor curve for quartz was refined to fit the entropies, densities, temperature, and pressure at the revised boiling point and triple point.
 We calculated the temperature and density structure in isentropically decompressing silica. For entropies below the critical point, the temperature along a release isentrope is strongly buffered at the point of intersection with the liquid-vapor curve. In addition, the change in sound speed upon entering the liquid side of the phase boundary leads to a plateau in the density profile at the state where the isentrope intersects the liquid-vapor curve. Then using a two-phase absorption model for silica, we predict the apparent temperature of the decompressing silica using a radiative transfer model.
 The predicted post-shock temperatures are in excellent qualitative agreement with the experimental data (Table 4). Using the calculated entropies in the shock state, quartz post-shock temperature data initially follow the liquid-vapor phase boundary and diverge as the entropy approaches the critical point in a manner predicted by the radiative transfer model. The experimental data are self-consistent with the independently-derived liquid-vapor curve. The new model critical point for silica isTCP = 5130 K, ρCP = 0.508 g cm−3, PCP = 0.13 GPa, and SCP = 5150 J kg−1 K−1.
 Planetary impact events commonly generate the shock pressures required for vaporization upon decompression (100's GPa). The revised critical shock pressures for vaporization are lower than previously estimated, primarily due to the revised entropy on the quartz Hugoniot (Table 6). Based on the systematics of silicate fluids, we expect similar revision of the Hugoniot entropies of major mantle minerals. Hence, shock-induced vaporization is a significant process during the end stage of planet formation and subsequent impact cratering events.
 RGK and DKS were supported by DOE NNSA SSGF program under grant DE-FC52-08NA28752. Funding was provided by G. W. Collins LLNL LDRD “Exploring Giant Planets”. We thank the staff at the Jupiter Laser Facility. We also thank Stephanie Uhlich and Walt Unites for technical support.