## 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, *P*_{1} ≈ 200 GPa (*1*) and *P*_{2} ≈ 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 *P*_{1}; 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 *P*_{2}. 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 Δ*P*_{IM} ≈ 95 GPa and a relative specific volume change of Δ*V*/*V*_{1} ≈ –0.2 while conserving entropy Δ*S* ≈ 0. Assuming the temperature is *T*_{1} at the start of the transition, the goal is to estimate *T*_{2} at *P*_{2}. 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 *T*_{2}/*T*_{1}. They provide two evaluations using parameters obtained from ab initio models: *C*_{P}, the specific heat at constant pressure in the metallic fluid (their equation 2, here called DKR-2), and *C*_{coex}, the specific heat in the coexisting two-phase fluid (their equation 3; DKR-3). The definition of specific heat *C _{X}* =

*dQ*/

*dT*|

*=*

_{X}*T dS*/

*dT*|

*describes the temperature change associated with heat transfer under constraint Δ*

_{X}*X*= 0; note that entropy is not conserved (Δ

*S*= Δ

*Q*/

*T*). To obtain a temperature drop, Desjarlais

*et al*. need Δ

*Q*< 0, so they extract the latent heat (–Δ

*H*

_{IM}) from the fluid (i.e., cooling) even though Δ

*S*= 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 Δ

*X*= Δ

*P*= 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*):

*C*

_{coex}=

*T*(

*dS*/

*dT*|

_{coex}) =

*C*

_{P}–

*T*(

*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 Δ

*X*= Δ

*V*= 0. The constraint Δ

*V*= 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 (Δ

*S*= –Δ

*H*

_{IM}/

*T*), either isobaric (Δ

*P*= 0, DKR-2) or isochoric (Δ

*V*= 0, DKR-3), in both cases transforming metal to insulator. There is no correspondence to the experiments where isentropic (Δ

*S*= 0) compression (Δ

*P*

_{IM}≈ 95 GPa and Δ

*V*/

*V*

_{1}≈ –0.2) transforms insulator to metal.

The compression path, sketched in Fig. 1, follows isentrope *S*_{1}^{I} to the coexistence line at (*P*_{1}, *T*_{1}), then enters the mixed-phase region and follows the coexistence line to (*P*_{2}, *T*_{2}) where the transformation is complete; further compression continues along *S*_{2}^{M}. Because the process is isentropic, *S*_{2}^{M} = *S*_{1}^{I}. The superscripts I and M refer to the insulating and metallic phases, respectively. Two other isentropes, *S*_{2}^{I} and *S*_{1}^{M}, intersect the coexistence line at (*P*_{2}, *T*_{2}) and (*P*_{1}, *T*_{1}), respectively. At (*P*_{1}, *T*_{1}) the phase transition can be accomplished by heating at constant *T* and *P* with heat energy Δ*Q*_{1} equal to the latent heat at *T*_{1}: Δ*Q*_{1} = Δ*H*_{IM}(*T*_{1}) = *T*_{1}Δ*S*_{IM}(*T*_{1}) where Δ*S*_{IM}(*T*_{1}) = *S*_{1}^{M} – *S*_{1}^{I}; in general, Δ*S*_{IM} is a function of *T*. Because *S*_{2}^{M} = *S*_{1}^{I}, it follows that Δ*S*_{IM}(*T*_{1}) = *S*_{1}^{M} – *S*_{2}^{M}, and Δ*S*_{IM}(*T*_{2}) = *S*_{1}^{I} – *S*_{2}^{I}.

We can calculate *T*_{2}/*T*_{1} along a path comprising a connected sequence of reversible thermodynamic process steps, α, spanning the IM transition and subject to the constraints ∑_{α }Δ*S*_{α} = 0 and ∑_{α }Δ*P*_{α} = Δ*P*_{IM}. 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 (*P*_{1}, *T*_{1}). 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 Δ*Q _{a}* = Δ

*H*

_{IM}(

*T*

_{1}) =

*T*

_{1}Δ

*S*

_{IM}(

*T*

_{1}) to transform from insulator to metal; therefore, Δ

*S*= +Δ

_{a}*S*

_{IM}(

*T*

_{1}), Δ

*P*= 0, and Δ

_{a}*T*= 0. Step

_{a}*b*is isentropic compression from (

*P*

_{1},

*T*) to (

_{a}*P*

_{2},

*T*) along

_{b}*S*

_{1}

^{M}in the pure metallic phase: Δ

*S*= 0 and Δ

_{b}*P*= 95 GPa. Finally, step

_{b}*c*is isobaric cooling (i.e., DKR-2) from (

*P*

_{2},

*T*,

_{b}*S*

_{1}

^{M}) to (

*P*

_{2},

*T*,

_{c}*S*

_{2}

^{M}): Δ

*S*= –Δ

_{c}*S*

_{IM}(

*T*

_{1}) and Δ

*P*= 0. By construction, ∑

_{c}_{α }Δ

*S*

_{α}= 0, ∑

_{α }Δ

*P*

_{α}= Δ

*P*

_{IM}, and insulator transforms to metal. After step

*a*,

*T*=

_{a}*T*

_{1}because Δ

*T*= 0; therefore,

_{a}*T*

_{2}/

*T*

_{1}= (

*T*/

_{b}*T*)(

_{a}*T*/

_{c}*T*). The DKR-2 calculation provides

_{b}*T*/

_{c}*T*≈ 0.83 (step

_{b}*c*). The remaining term,

*T*/

_{b}*T*, can be determined from the slope of the isentrope in the pure metallic phase. From basic thermodynamic principles,

_{a}*T dP*/

*dT*|

_{S}=

*B*

_{S}/γ, where

*B*

_{S}is the isentropic bulk modulus and γ is the Grüneisen parameter. Integration along the isentrope from (

*P*

_{1},

*T*) to (

_{a}*P*

_{2},

*T*) leads to

_{b}*et al*.); therefore,

*T*<

_{b}*T*. We estimate γ ≈ –1.2 by examining the isentropes plotted in figure 1 of (

_{a}*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

*B*

_{S}≈ 525 GPa [interpolated from Pierleoni

*et al*. (

*5*) supplementary table 5 near 240 GPa, midway along the isentrope between (

*P*

_{1},

*T*) and (

_{a}*P*

_{2},

*T*)] and γ ≈ –1.2, we find

_{b}*T*/

_{b}*T*≈ 0.80; thus,

_{a}*T*

_{2}/

*T*

_{1}= (

*T*/

_{b}*T*)(

_{a}*T*/

_{c}*T*) ≈ 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 (

_{b}*P*

_{1},

*T*

_{1}) to reduce the entropy by –Δ

*S*

_{IM}(

*T*

_{2}); b*, isentropic compression along

*S*

_{2}

^{I}from

*P*

_{1}to (

*P*

_{2},

*T*

_{2},

*S*

_{2}

^{I}); and c*, isothermal and isobaric heating by +Δ

*H*

_{IM}(

*T*

_{2}) =

*T*

_{2}Δ

*S*

_{IM}(

*T*

_{2}) to transform from insulator to metal, reaching (

*P*

_{2},

*T*

_{2},

*S*

_{2}

^{M}).]

*T*_{2}/*T*_{1} can also be calculated by direct integration of the Clausius-Clapeyron equation along the coexistence line:*T*_{1}, *P*_{1}) and terminates at (*T*_{2}, *P*_{2}). Here, Δ*V*_{IM}(*P*) and Δ*H*_{IM}(*P*) are the volume change and latent heat as a function of pressure along the coexistence line. From this equation,*H*_{IM} ≈ 2.62 kJ/g from Pierleoni *et al*. (*5*), a 3% volume discontinuity Δ*V*_{IM} ≈ –0.015 cm^{3}/g estimated from several studies (*5*–*8*), and from the experiments (*P*_{2} – *P*_{1}) ≈ 95 GPa, we find *T*_{2}/*T*_{1} ≈ 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 Δ*T* = *T*_{2} – *T*_{1} = *fT*_{1}, where *f* = –0.49 ± 0.16; therefore, *T*_{2}/*T*_{1} = *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, *B*_{S}, and *C*_{P} near the IM transition, as well as Δ*V*_{IM}(*P*) and Δ*H*_{IM}(*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.