Technical Comments

Response to Comment on “Insulator-metal transition in dense fluid deuterium”

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Science  22 Mar 2019:
Vol. 363, Issue 6433, eaaw1970
DOI: 10.1126/science.aaw1970

Abstract

In their comment, Desjarlais et al. claim that a small temperature drop occurs after isentropic compression of fluid deuterium through the first-order insulator-metal transition. We show that their calculations do not correspond to the experimental thermodynamic path, and that thermodynamic integrations with parameters from first-principles calculations produce results in agreement with our original estimate of the temperature drop.

The recent experiments reported in Celliers et al. (1) and earlier experiments of Knudson et al. (2) compressed samples of liquid deuterium to higher than 300 GPa using quasi-isentropic compression methods. Optical reflectance signals from the two experiments show distinct transitions at two different pressures, P1 ≈ 200 GPa (1) and P2 ≈ 300 GPa (2). According to our interpretation of both experiments, in (1) the samples were inertially confined because of short time scales and the optical reflectance tracked the initial stages of the first-order insulator-metal (IM) transition at pressure P1; in (2), much longer time scales and lateral gradients resulted in turbulent mixing, which suppressed the optical reflectance signal until the IM transition was complete, at pressure P2. Therefore, we think the two experiments identify the pressures at the start and completion of the IM transition, respectively (i.e., the pressure extent over which the isentrope passes through the coexistence region). The transformation observed in the experiments spans a pressure change of ΔPIM ≈ 95 GPa and a relative specific volume change of ΔV/V1 ≈ –0.2 while conserving entropy ΔS ≈ 0. Assuming the temperature is T1 at the start of the transition, the goal is to estimate T2 at P2. First, we review the method used by Desjarlais et al. (3), then follow with a detailed thermodynamic analysis.

Desjarlais et al. use specific heat to estimate T2/T1. They provide two evaluations using parameters obtained from ab initio models: CP, the specific heat at constant pressure in the metallic fluid (their equation 2, here called DKR-2), and Ccoex, the specific heat in the coexisting two-phase fluid (their equation 3; DKR-3). The definition of specific heat CX = dQ/dT|X = T dS/dT|X describes the temperature change associated with heat transfer under constraint ΔX = 0; note that entropy is not conserved (ΔS = ΔQ/T). To obtain a temperature drop, Desjarlais et al. need ΔQ < 0, so they extract the latent heat (–ΔHIM) from the fluid (i.e., cooling) even though ΔS = 0 in the experiments. As justification, Desjarlais et al. state: “Traversal of the phase boundary isothermally … results in an increase in entropy”; however, there can be no increase in entropy during isentropic compression, and isothermal traversal cannot occur in the experiments. DKR-2 constrains ΔX = ΔP = 0, whereas DKR-3 constrains cooling to be along the coexistence line. Using the Maxwell relations, we can reduce DKR-3 to equation 4.19 in (4): Ccoex = T(dS/dT|coex) = CPT(dV/dT|P)(dP/dT|coex), which gives the specific heat of a substance in two-phase coexistence at constant V, such as liquid water and its vapor in a sealed container; that is, the constraint is ΔX = ΔV = 0. The constraint ΔV = 0 causes P to increase as T decreases, like the experiments, but cooling transforms metal to insulator, contrary to the experiments. To recapitulate: Desjarlais et al. calculate cooling (ΔS = –ΔHIM/T), either isobaric (ΔP = 0, DKR-2) or isochoric (ΔV = 0, DKR-3), in both cases transforming metal to insulator. There is no correspondence to the experiments where isentropic (ΔS = 0) compression (ΔPIM ≈ 95 GPa and ΔV/V1 ≈ –0.2) transforms insulator to metal.

The compression path, sketched in Fig. 1, follows isentrope S1I to the coexistence line at (P1, T1), then enters the mixed-phase region and follows the coexistence line to (P2, T2) where the transformation is complete; further compression continues along S2M. Because the process is isentropic, S2M = S1I. The superscripts I and M refer to the insulating and metallic phases, respectively. Two other isentropes, S2I and S1M, intersect the coexistence line at (P2, T2) and (P1, T1), respectively. At (P1, T1) the phase transition can be accomplished by heating at constant T and P with heat energy ΔQ1 equal to the latent heat at T1: ΔQ1 = ΔHIM(T1) = T1ΔSIM(T1) where ΔSIM(T1) = S1MS1I; in general, ΔSIM is a function of T. Because S2M = S1I, it follows that ΔSIM(T1) = S1MS2M, and ΔSIM(T2) = S1IS2I.

Fig. 1 Thermodynamic paths for compression.

The upper and lower frames show thermodynamic paths on the T-P and S-P planes, respectively; both frames share the same P scale. The experimental compression path (red solid line segments) enters the mixed-phase coexistence region at (P1, T1, S1I) and follows the coexistence line at equilibrium (blue chain-dashed curve in upper frame) until the transformation is complete at (P2, T2, S2M = S1I). For T calculations, an alternate path (purple dashed curves) consists of step a, isobaric and isothermal heating by +ΔHIM(T1) = T1ΔSIM(T1) to transform from insulator (P1, T1, S1I) to metal (P1, T1, S1M); step b, isentropic compression along S1M from (P1, Ta = T1, S1M) to (P2, Tb, S1M); and step c, isobaric cooling from Tb to Tc to reach (P2, Tc = T2, S2M = S1I). Step c corresponds to Desjarlais et al. Eq. 2 (DKR-2). Desjarlais et al. Eq. 3 (DKR-3) follows an isochore in the coexistence region and drives the transition from metal to insulator, contrary to the experiments (see text). Neither DKR-2 nor DKR-3 corresponds to the experimental path.

We can calculate T2/T1 along a path comprising a connected sequence of reversible thermodynamic process steps, α, spanning the IM transition and subject to the constraints ∑α ΔSα = 0 and ∑α ΔPα = ΔPIM. One possibility is sketched in Fig. 1, on both the T-P and S-P planes: three steps starting with 100% fraction of the insulating phase at (P1, T1). We label each step, and its T at completion, with the subscripts a, b, and c, respectively. Step a is isobaric and isothermal heating by the latent heat ΔQa = ΔHIM(T1) = T1ΔSIM(T1) to transform from insulator to metal; therefore, ΔSa = +ΔSIM(T1), ΔPa = 0, and ΔTa = 0. Step b is isentropic compression from (P1, Ta) to (P2, Tb) along S1M in the pure metallic phase: ΔSb = 0 and ΔPb = 95 GPa. Finally, step c is isobaric cooling (i.e., DKR-2) from (P2, Tb, S1M) to (P2, Tc, S2M): ΔSc = –ΔSIM(T1) and ΔPc = 0. By construction, ∑α ΔSα = 0, ∑α ΔPα = ΔPIM, and insulator transforms to metal. After step a, Ta = T1 because ΔTa = 0; therefore, T2/T1 = (Tb/Ta)(Tc/Tb). The DKR-2 calculation provides Tc/Tb ≈ 0.83 (step c). The remaining term, Tb/Ta, can be determined from the slope of the isentrope in the pure metallic phase. From basic thermodynamic principles, T dP/dT|S = BS/γ, where BS is the isentropic bulk modulus and γ is the Grüneisen parameter. Integration along the isentrope from (P1, Ta) to (P2, Tb) leads toTaTbdT/T=P1P2dPγ/BS(1)orTb/Ta=exp[P1P2dPγ/BS](2)The slope is negative because γ < 0 (as noted in Desjarlais et al.); therefore, Tb < Ta. We estimate γ ≈ –1.2 by examining the isentropes plotted in figure 1 of (2); note that the value γ = –2.3 from Desjarlais et al. does not apply because it was evaluated within the coexistence region, not the pure metallic phase. Substituting BS ≈ 525 GPa [interpolated from Pierleoni et al. (5) supplementary table 5 near 240 GPa, midway along the isentrope between (P1, Ta) and (P2, Tb)] and γ ≈ –1.2, we find Tb/Ta ≈ 0.80; thus, T2/T1 = (Tb/Ta)(Tc/Tb) ≈ 0.80 × 0.83 ≈ 0.66. The path chosen for the calculation is not unique. [A similar calculation on the insulating side of the transition is as follows: a*, isobaric cooling from (P1, T1) to reduce the entropy by –ΔSIM(T2); b*, isentropic compression along S2I from P1 to (P2, T2, S2I); and c*, isothermal and isobaric heating by +ΔHIM(T2) = T2ΔSIM(T2) to transform from insulator to metal, reaching (P2, T2, S2M).]

T2/T1 can also be calculated by direct integration of the Clausius-Clapeyron equation along the coexistence line:T1T2dT/T=P1P2dPΔVIM(P)/ΔHIM(P)(3)where the integration starts at (T1, P1) and terminates at (T2, P2). Here, ΔVIM(P) and ΔHIM(P) are the volume change and latent heat as a function of pressure along the coexistence line. From this equation,T2/T1=exp[P1P2dPΔVIM(P)/ΔHIM(P)](4)Using estimates for ΔHIM ≈ 2.62 kJ/g from Pierleoni et al. (5), a 3% volume discontinuity ΔVIM ≈ –0.015 cm3/g estimated from several studies (58), and from the experiments (P2P1) ≈ 95 GPa, we find T2/T1 ≈ 0.58. The original estimate in Celliers et al. (1), based on a finite difference evaluation of the Clausius-Clapeyron equation, resulted in a scaling factor for ΔT = T2T1 = fT1, where f = –0.49 ± 0.16; therefore, T2/T1 = f + 1 = 0.51 ± 0.16. Our study (1) also accounted for cooling from the aluminum piston in (2), which, combined with long time scales, turbulent mixing, and convective heat exchange, might account for a large temperature drop; however, quantitative calculations found a small effect (~100 K) because the heat capacity of the aluminum piston is much lower than that of the deuterium fluid. Thus, the experimental path is isentropic to a good approximation, and to simplify the discussion in this response we considered purely isentropic processes. The two new estimates given here agree within the uncertainty stated in (1).

To summarize: Thermodynamic path integration is in reasonable agreement with Clausius-Clapeyron integration; all estimates are within the uncertainty range quoted in (1); and finally, anomalous specific heat is not required to explain the temperature drop, contrary to the conclusion of Desjarlais et al. Key to these calculations are accurate values for γ, BS, and CP near the IM transition, as well as ΔVIM(P) and ΔHIM(P).

References and Notes

Acknowledgments: This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.
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