Volume 50, Issue 18 e2023GL103994
Research Letter
Open Access

Thermal Conductivity of Hydrogen at High Pressure and High Temperature: Implications to Giant Planets

Sean R. Shieh

Corresponding Author

Sean R. Shieh

Department of Earth Sciences, University of Western Ontario, London, ON, Canada

Department of Physics and Astronomy, University of Western Ontario, London, ON, Canada

Correspondence to:

S. R. Shieh and W.-P. Hsieh,

[email protected];

[email protected]

Contribution: Conceptualization, ​Investigation, Writing - original draft, Funding acquisition

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Wen-Pin Hsieh

Corresponding Author

Wen-Pin Hsieh

Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan

Department of Geosciences, National Taiwan University, Taipei, Taiwan

Correspondence to:

S. R. Shieh and W.-P. Hsieh,

[email protected];

[email protected]

Contribution: Conceptualization, Methodology, ​Investigation, Writing - original draft, Funding acquisition

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Yi-Chih Tsao

Yi-Chih Tsao

Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan

Contribution: Methodology

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First published: 21 September 2023

Abstract

Hydrogen (H2) is the most abundant constituent in giant planets, but its thermal conductivity Λ under extreme pressure-temperature (P-T) conditions remains largely unknown. Here we report the Λ of H2 from ambient to 60.2 GPa at 300 K and from 300 to 773 K at 2.1 GPa. At 300 K, the Λ of liquid H2 fluctuates at ∼0.7–1.1 W m−1 K−1. Upon crystallization to H2-I phase, the Λ jumps to 5.5 W m−1 K−1 at 7.2 GPa, and monotonically increases with pressure to ∼27 W m−1 K−1 at 60.2 GPa. Upon heating, the Λ of liquid H2 at 2.1 GPa scales with T0.68. Moreover, the density (ρ)-dependent compressional sound velocity (Vp) of liquid and solid H2 derived from Brillouin frequency data both follow the Birch's law. Besides the novel insights into the physics of thermal transport in H2 under extreme conditions, our results significantly advance the modeling of Λ-Vp-ρ relationship in a planet with H2.

Key Points

  • Thermal conductivity of liquid H2 varies in the range of 0.7–1.1 W m−1 K−1 at high pressures and room temperature

  • Thermal conductivity of solid H2-I increases from 5.5 W m−1 K−1 at 7.2 GPa to 27 W m−1 K−1 at 60.2 GPa

  • Our model suggests that the thermal conductivity of liquid H2-He mixture is in the range of 0.70–1.0 W m−1 K−1 at 300K

Plain Language Summary

Hydrogen is the most abundant element in the universe and is also the major constituent in the giant plants (Jupiter, Saturn, Uranus and Neptune) in the solar system. Although most of the surface temperatures of those giant planets are colder than Earth's, their interior temperatures are actually able to reach several thousand degrees Kelvin. The heat flows within these giant planets are very active, while knowledge of heat conduction and propagation are largely unknown. This study investigates the high-pressure thermal conductivity of H2 at room temperature and high temperature conditions. Our results show that the liquid H2 exhibits low thermal conductivity, in the range of 0.7–1.1 W m−1 K−1 at high pressures and room temperature. However, appearance of solid H2 will increase thermal conductivity significantly and reach ∼27 W m−1 K−1 at 60.2 gigapascals and room temperature. Based on our model, the low thermal conductivity of liquid H2-He mixture may suppress the heat loss of the giant planets and explain why their surfaces are cold, but interiors are hot.

1 Introduction

Hydrogen (H2) is the most abundant element and essential constituent in the universe. H2 is also the major component in giant planets in the solar system. For example, molecular H2 is believed to exist at the upper parts of the gas and ice planets such as Jupiter, Saturn, Uranus and Neptune (Arridge et al., 2014; DeMarcus, 1958; Nettelmann et al., 2008). Furthermore, Jupiter and Saturn were suggested to be made of mostly hydrogen and helium, about 75%–85% of their total masses (Helled et al., 2020) and it's evidenced by their average densities of 1.326–1.327 and 0.678–0.688 g cm−3, respectively (Guillot, 2005; Helled et al., 2020). Similarly, the average densities of Uranus and Neptune are also reported in the range of 1.27–1.64 g cm−3 (Guillot, 2005). Furthermore, surface temperatures for Jupiter, Saturn, Uranus, and Neptune at 1 bar are reported to be about 165, 135, 75 and 70 K, respectively (Guillot, 2005; Helled et al., 2020), which are indeed colder than the Earth. On the other hand, internal temperatures of Uranus and Neptune were estimated to be about 1774–2000 K at depths corresponding to ∼10 GPa; similarly, those of Jupiter and Saturn may reach about 3000–4000 K at depths corresponding to ∼10 GPa and ∼5500–6800 K at depths corresponding to 200 GPa (Arridge et al., 2014; Guillot, 2005; Helled et al., 2020; Nettelmann et al., 2013). These estimates suggest that the giant planets possess with very high internal energy but the heat transfer and propagation by major constituents, such as H2, at various states are largely unknown. Not to mention, physical properties of hydrogen are of great importance to model the thermal and structural evolution of giant planets in the solar system and other exoplanetary bodies. In fact, thermal conductivity of important planetary phases plays a central role in controlling the heat conduction and thermal flow within the planets and shaping the outmost features of the planets. Nevertheless, limited thermal conductivity of common volatile phases, such as H2O, CO2, He, Ar, and methane were reported to date (Chen et al., 2011; Hsieh et al., 2022; Meyer et al., 2022; Shieh et al., 2022). Importantly, H2 exhibits distinct behaviors under high pressure (P) and high temperature (T) conditions. At 300 K, gaseous H2 can be liquefied at P > 35 MPa and liquid H2 is stable to ∼5.4 GPa (Mao & Hemley, 1994). Upon further compression, the liquid H2 transforms to a solid phase I with a hexagonal close-packed (hcp) structure that is reported to be stable to ∼180 GPa (Mao & Hemley, 1994). Both liquid and solid phase I are regarded as molecular phases, but are expected to show different thermal conduction capability. Note that conventional “molecular phase” are sometimes used and reported as one term without distinction between liquid and solid on many occasions to describe the upper layer structure of giant planets and exoplanets (Arridge et al., 2014; DeMarcus, 1958; Helled et al., 2020). The sound wave velocity of H2 may play a key role for understanding the internal structures of the giant planets, but knowledge of the sound wave velocity of H2 is very limited (Zha et al., 1993). Here we report the thermal conductivity and compressional velocity of liquid and solid H2 under a variety of P-T conditions, and discuss the physical mechanisms of thermal transport within such quantum system, along with their effects on the thermal conduction and evolution in giant planets that contain H2.

2 Methods

To perform high-pressure, room-temperature thermal conductivity measurements on H2, we first thermally evaporated ∼90 nm-thick Al film thermal transducer on an ∼15 μm-thick muscovite mica sheet that serves as a reference substrate. We then loaded the mica into a symmetric, 300 μm-culet diamond-anvil cell (DAC), where a Re gasket was used. We compressed the mica substrate by high-pressure gas loading H2 with a purity of 99.9999% that serves as the pressure medium and the material of interest. We measured the pressure within the DAC's sample chamber by characterizing the pressure-induced shift of the fluorescence or Raman spectrum of the ruby spheres (Dewaele et al., 2004), with a typical uncertainty of <5%, depending on the pressure range. Note that when the pressure is increased to P > 40 GPa, we also measured the pressure inferred from the Raman spectrum of the diamond anvil (Akahama & Kawamura, 2004) and compared with pressure determined from ruby fluorescence method. Our results showed that the difference between these two probes is typically <2 GPa, confirming the reliability of the reported pressure. The present experimental setup and sample geometry are essentially the same as previous reports (see Hsieh et al., 2009; Hsieh, 20152021 for more details).

For thermal conductivity measurements at simultaneous high P-T conditions, we employed an externally-heated DAC (EHDAC) (Hsieh, 2021; Lai et al., 2020) to generate such extreme environments. Note that our EHDAC is equipped with a gas membrane, which enables in situ tuning of the pressure within the EHDAC when the temperature is varied, thus allowing us to measure the temperature-dependent thermal conductivity of H2 at a fixed pressure, for example, 2.1 GPa and up to ∼773 K. The assemblage and temperature measurement of the EHDAC in this study both follow previous reports (Hsieh, 2021; Lai et al., 2020).

We used time-domain thermoreflectance (TDTR) to measure the lattice thermal conductivity of H2 at high pressure and a wide range of temperature. TDTR is a well-developed ultrafast optical pump-probe method that has been widely applied to measure the thermal conductivity of materials at extremely high pressures (Hsieh et al., 2020; Zhou et al., 2022). High-pressure lattice thermal conductivity of H2 was obtained by comparing the ratio –Vin/Vout as a function of delay time between pump and probe beams with calculations using a bi-directional heat diffusion model that considers heat flux into both the H2 and the mica substrate (Schmidt et al., 2008). Example data along with heat diffusion model calculations are shown in Figure S1 in Supporting Information S1. The heat diffusion model contains several input parameters, including laser spot size (≈7.6 μm in radius) and thickness, thermal conductivity, and volumetric heat capacity of each layer (i.e., H2, Al film, and mica substrate), but the H2's thermal conductivity is the only significant unknown and free parameter to be determined in our data analysis.

To derive the H2's thermal conductivity, its volumetric heat capacity is a critical parameter. The volumetric heat capacity of liquid H2 at room temperature and high pressures was taken from Mills et al. (1977) and Matsuishi et al. (2002). At P > 5.4 GPa, the pressure-dependent volumetric heat capacity of solid H2-I was obtained by multiplying the molar heat capacity fitted into a polynomial by the molecular density (both were taken from Zha et al. (1993)). Note that the uncertainty in the derived thermal conductivity of H2 is mainly from the data analysis, not from the measurements. We estimated that the uncertainties in all the parameters used in our heat diffusion model would propagate ≈15% error in the derived thermal conductivity of H2 before 20 GPa, and less than 25% error at ≈20–70 GPa. More details of the data analysis and uncertainty evaluations are given in Figure S2 in Supporting Information S1 and related literature (Hsieh, 2021; Hsieh et al., 2009).

Besides the thermal conductivity, we have also measured the Brillouin frequency of H2 at room temperature and high pressures using time-domain stimulated Brillouin scattering (Hsieh et al., 2009), which is an inelastic light scattering from the acoustic vibrational modes of a material and correlated with its elastic constants. We used a backscattering geometry where the incident and reflected optical beams are nearly normal to the surface of the sample. As a result, for the longitudinal modes, the Brillouin frequency fB = 2NVp/λ, where N is the index of refraction of H2, Vp the compressional (longitudinal) sound velocity, and λ the laser wavelength (785 nm). The uncertainty in our Brillouin frequency measurements is typically ≈1%.

3 Results and Discussion

Figure 1a presents the thermal conductivity Λ of H2 at room temperature up to ∼60 GPa. In the liquid phase, the Λ is weakly dependent on pressure, fluctuating between 0.7 and 1.1 W m−1 K−1, followed by a decrease of ∼40% right before the pressure-induced liquid-solid transition. This intriguing behavior could be due to the less effective lattice vibrations in the liquid phase near the transition pressure, or a change in heat capacity around phase transition. Further experimental and computational studies are required to understand the detailed mechanisms. Above ∼5.4 GPa, the H2 solidifies as molecular solid phase I. The Λ of H2-I phase is 5.5 W m−1 K−1 at 7.2 GPa, indicating approximately an order-of-magnitude increase across the liquid-solid transition, and increases with pressure to 27 W m−1 K−1 at 60.2 GPa. If we assume the Λ of H2-I phase can be phenomenologically modeled with pressure as ΛH2(P) = αPn, where α is a normalization constant, the obtained pressure slope n is determined as 0.67(±0.04) by the linear regression in the lnΛ-lnP plot. Note that the pressure slope of H2-I phase is slightly smaller than that of the solid He (n = 0.72) (Hsieh et al., 2022), likely due to the smaller anharmonicity in the H2-I phase.

Details are in the caption following the image

(a) Thermal conductivity of liquid (L, red squares) and solid (S, black circles) H2 at high pressure and room temperature. The data for each phase include several measurement runs that yield consistent results. In the liquid phase, the thermal conductivity fluctuates around 0.8 W m−1 K−1, while in the solid phase, the thermal conductivity increases with pressure with a slope n of 0.67, slightly smaller than that of the solid He (open blue triangles, n = 0.72) (Hsieh et al., 2022). The black solid line is the fitting to the thermal conductivity of solid H2. The black dashed curve shows the predicted thermal conductivity of solid H2 based on the LS eq (see text). The data uncertainty, which is mainly caused by data analysis, is typically ≈15% before 20 GPa and ≈25% at 20–60 GPa. The vertical black dashed and blue dotted lines represent the pressure-induced phase transitions from liquid to solid phase for H2 around 5.4 GPa and for He around 12 GPa, respectively. (b) Temperature dependence of the thermal conductivity of liquid H2 obtained at 2.1 GPa. The thermal conductivity scales with Tm, where m = 0.68(±0.18).

We further studied the Λ of liquid H2 at simultaneous high P-T conditions (2.1 GPa and 300–773 K) to understand its temperature dependence at a given high pressure (Figure 1b). Although our measurement temperature was only up to 773 K, Λ of liquid H2 shows a positive relation with temperature. Similar behavior was also found for liquid He (Hsieh et al., 2022). Such positive temperature dependence is likely that in liquid, the thermal energy transport is typically made by diffusion of heat from atom to atom. As temperature increases, the vibration of atoms becomes stronger, which enhances the heat capacity and efficiency of energy transfer between atoms, leading to a higher thermal conductivity. Interestingly, the positive temperature dependence of both liquid H2 and He is opposite to that of typical dielectric crystals (Roufosse & Klemens, 1973) and molecular solid Ar (Hsieh et al., 2022), in which when the applied temperature is higher than their Debye temperatures, their Λ decreases with increasing temperature. To quantify the temperature dependence of Λ of liquid H2, we assumed the Λ of liquid H2 scales with Tm, and obtained the linear regression slope in the lnΛ-lnT plot as m = 0.68(±0.18).

It's important to point out that the Leibfried-Schlömann (LS) equation has often been used to predict how the Λ of a dielectric crystal changes with unit cell volume (typically via the application of pressure) and temperature (Roufosse & Klemens, 1973; Slack, 1979):
urn:x-wiley:00948276:media:grl66376:grl66376-math-0001(1)
where V is volume, ωD is Debye frequency, γ is Grüneisen parameter, T is temperature, and A is a normalization constant. As a simple physical model, the LS equation was originally formulated for isotropic dielectric crystals, where the thermal energy transport was assumed to be predominantly controlled by anharmonic three phonon scatterings among acoustic phonons. It has been shown that for molecular solid Ar (Goncharov et al., 2012), H2O ice VII (Chen et al., 2011), NaCl (Hsieh et al., 2022), and MgO (Dalton et al., 2013), their Λ as a function of pressure is in reasonable agreement with that predicted by the LS equation. However, for solid He, compared to the experimental data, the LS equation overpredicts the Λ at high pressures, for example, an overprediction of ∼42% at 20.5 GPa and of ∼78% at 50 GPa (Hsieh et al., 2022).

Our present results (Figure 1a) are used to examine the applicability of LS equation to predict the Λ(P) of solid H2, which could provide insights into the mechanisms of heat and phonon transport in such quantum system. To this end, for simplicity, the Grüneisen parameter, elastic anisotropy, and Poisson ratio are all assumed to be weakly dependent on pressure, allowing Λ to be simplified to Λ = B V5/6 KT3/2 under a given temperature, where B is a normalization factor determined by fitting the thermal conductivity data for solid H2 at 7.2 GPa and KT the isothermal bulk modulus. With the equation of state and elastic constants adopted from Loubeyre et al. (1996), the LS equation predicts a Λ(P) (black dashed curve in Figure 1a) with a much larger increasing rate with pressure than our experimental data. The predictions of the Λ(P) of solid H2 deviated strongly with pressure, suggesting an overprediction of ∼31% at 12.5 GPa and of ∼100% (factor of 2) at P > 23 GPa, where the overprediction is much larger than our data uncertainties. Similar overpredictions on the Λ(P) of solid He were also reported as we discussed above (Hsieh et al., 2022). We note that since the Grüneisen parameter typically decreases with increasing pressure, such overprediction by the LS equation is thus expected to be further enhanced if the pressure dependence of Grüneisen parameter is taken into account. Our experimental findings show that the LS equation predicts higher Λ(P) values with increasing pressure, suggesting that in addition to the typical anharmonic three phonon scattering process within the acoustic branches, other phonon scattering processes and/or quantum effects may effectively reduce the heat transport at higher pressures. Future experimental and computational studies are required to better understand the fundamental phonon transport mechanisms in the molecular solid H2-I phase under extreme pressures.

In addition to the LS equation, under the Debye approximation (Dalton et al., 2013; de Koker, 2010), thermal conductivity of materials has also been phenomenologically modeled as a function of density ρ and temperature T:
urn:x-wiley:00948276:media:grl66376:grl66376-math-0002(2)
where Λ0, ρ0, and T0 are the reference thermal conductivity, molecular density, and temperature, respectively. Equation 2 enables simple scaling of the thermal conductivity with molecular density and temperature, along with applications to understand the thermal properties of Earth and planetary interiors based on geophysical observations. The density exponent g can be expressed as 3γ + 2q − 1/3 (Dalton et al., 2013; de Koker, 2010), where the Grüneisen parameter γ is defined as (∂lnν/∂lnρ)T, ν is the phonon frequency, and q is defined as −(∂lnγ/∂lnρ)T. The temperature exponent m is assumed to be −0.68 for liquid H2 (Figure 1b, assuming it is weakly dependent on the pressure). The g value for solid H2 was obtained by the linear slope in its ln(Λ/Λ0)-ln(ρ/ρ0) plot, where the Λ0 and ρ0 are the thermal conductivity (Figure 1a) and density (taken from Loubeyre et al., 1996) at 7.2 GPa. We obtained g = 1.74(±0.11) for the solid H2 (Figure 2), smaller than that of the solid He with g = 2.33 (Hsieh et al., 2022), presumably due to the more harmonic interatomic potentials. For the liquid phase, except for the data that decrease by ∼40% as approaching to the phase boundary, the thermal conductivity is weakly dependent on the density with a slope of g = 0.2(±0.29) (Figure 2), suggesting a small anharmonicity that is insensitive to the density change for the interatomic potential of liquid H2.
Details are in the caption following the image

Logarithmic normalized thermal conductivity as a function of logarithmic normalized density for liquid H2 (red squares) with g = 0.2, solid H2 (black circles) with g = 1.74, and solid He (blue open triangles) with g = 2.33 (Hsieh et al., 2022).

Figure 3a shows the pressure dependence of the Brillouin frequency fB of H2 up to 60.2 GPa at room temperature. At 1 GPa the fB is 16.3 GHz and reaches 29.4 GHz at ∼5.1 GPa for the liquid phase (red squares). As the H2 solidifies at ∼5.4 GPa, the fB sharply increases by ∼30%, and eventually reaches 104 GHz at 60.2 GPa. The corresponding Vp (inset of Figure 3a) was derived by fB = 2NVp/λ, where the N was taken from Dewaele et al. (2003) assuming the N at 632.8 nm is the same as at 785 nm, and N at P > 35 GPa was extrapolated following the results in Birch (1961b). Again, across the liquid-solid transition, the Vp jumps up by nearly 30% and increases rapidly with pressure to ∼21.1 km s−1 at 60.2 GPa. Our results for the Vp of H2 are in an excellent agreement with previous data measured from conventional Brillouin scattering spectroscopy (Zha et al., 1993), indicating reliable Vp data extended to 60.2 GPa from this study. Note that we not only provide higher pressure data of Vp for H2 but also validate the Vp to 60 GPa. If the sound wave velocity of giant planets can be detected in the future, our obtained Vp of H2 would be of great help for understanding the internal structure and composition of giant planets.

Details are in the caption following the image

(a) Pressure dependence of the Brillouin frequency for liquid (red squares) and solid (black circles) H2 at room temperature. The inset shows the compressional sound velocity (Vp) versus pressure. Literature Vp data from Zha et al. (1993) are plotted as open triangles for comparison. The vertical dashed lines indicate the phase boundary between liquid (L) and solid (S) phase. (b) Vp of H2 as a function of density ρ. Both the Vp of liquid (red squares) and solid (black circles) H2 scale approximately linearly with the density, obeying the Birch's law, with a similar slope of ∼32. Literature data (Meyer et al., 2022) for other volatiles, including solid CH4 (green line) with a slope of 13.2 and solid H2O (blue line) with a slope of 6.1, are plotted for comparison.

Density dependence of Vp for liquid and solid H2 (red squares and black circles in Figure 3b, respectively) allows us to test the applicability of Birch's law to H2 over the density range we studied. The Birch's law (Birch, 1961a1961b) predicts that within a certain range of density ρ of a material, its Vp is linearly proportional to ρ: Vp =  + b, where a is a constant and b is related to the average atomic weight of the material. The Birch's law has been widely employed to project the radial profile of sound velocity in Earth and planetary interiors if the radial profile of density is known (or vice versa), providing critical insights into their internal structure and composition (Birch, 1961a; Liebermann & Ringwood, 1973). Here we found that both the Vp of liquid and solid H2 scale approximately linearly with their density, with essentially the same slope a = 32(±0.8), that is, both following the Birch's law. Though the Birch's law is typically found to describe a linear Vp-ρ dependence for a solid, our results suggest that it is also applicable to the liquid H2, which could enable better modeling of the Vp profile in a planet with liquid H2, if the density profile is known (or vice versa). Even if the lattice dynamics and interatomic potential that determine the elasticity and sound velocity (Chung, 1972) of liquid and solid H2 are different, their compressional velocities remain in the linear regime with respect to the wide range of density and compression we explored.

4 Implications for Giant Planets

It's believed that H2 and He mixtures (with He ∼18–25 wt%) can be found at the top layers of Saturn and Jupiter (McMahon et al., 2012). In addition, a previous study of the internal structure of Uranus suggested that the molecular H2–He mixture coexists with H2O liquid in different depths (Arridge et al., 2014). Even though our measured thermal conductivity of H2 differs from the reported pressure and temperature conditions in Uranus and other ice/gas giant planets, the ≥7 times increase of thermal conductivity across liquid-solid transition is first reported for H2. This means that solid H2 will conduct heat more effectively than liquid H2. At room temperature H2 exhibits as a liquid phase at pressure to ∼5.4 GPa, which is much higher than the liquid-solid transition pressure for H2O and CO2 phases. At room temperature liquid phase of H2O and CO2 is stable at pressure up to 0.8 and 0.6 GPa, respectively (Chen et al., 2011; Shieh et al., 2022). This suggests that the H2 liquid may be coexisting with H2O and CO2 ice at pressure to ∼5.4 GPa. Furthermore, the thermal conductivity of H2O ice is about 2.5 times higher than that of liquid H2 at 1.1 GPa for phase VI and increases to ∼3 times higher at about 2.1 GPa for phase VII. At 5.1 GPa, the thermal conductivity of H2O ice VII is about 8 times higher than that of liquid H2, suggesting a profound heat conduction gap between these two phases. Once the solid H2-I phase is formed, the comparable thermal conductivity of H2 and H2O is found at about 5.8–6.8 GPa, and subsequently their thermal conductivity gradually deviates from each other due to the steep trend of H2O ice VII. Comparing with He, the thermal conductivity of He has a slightly lower value compared to H2 below 4 GPa, but at >4 GPa it shows a slightly higher value than that of H2 liquid. However, once the H2-I phase is formed, the thermal conductivity of H2 jumps about three to four times higher than that of He. For the thermal conductivity of CO2, it also shows a slightly lower value than that of H2 liquid, except near 4.5–5.4 GPa, but becomes much smaller (about <4.5 times) than that of H2-I phase with pressure.

To understand the thermal conductivity of the top layers of gas planets, we calculated the H2-He mixture in the range of 70–90 vol% of H2 (Figure 4). Since the demixing of H2-He has been reported by previous studies (Guillot, 2005; Helled et al., 2020), we thus use a linear relationship for H2 and He individually to model the thermal conductivity of H2-He mixture. When both H2 and He are in the liquid phases, the thermal conductivity of H2-He mixture is in the range of 0.70–1.0 W m−1 K−1. When the H2-I phase appears but He remains in the liquid phase, the thermal conductivity of H2-He mixture increases to 4.3–7.2 W m−1 K−1. When both H2 and He are solid phases, the thermal conductivity of H2-He mixture would increase to about 6.2–7.2 W m−1 K−1 at 12 GPa and reaches 19.0–22.0 W m−1 K−1 at 55 GPa.

Details are in the caption following the image

Comparison of thermal conductivity of H2, H2O, He and CO2 at high pressure and room temperature. The dashed lines represent liquid phases and solid lines are solid phases. H290He10 stands for a mixture of ratio of 90%H2 to 10%He, H280He20 stands for a mixture of ratio of 80%H2 to 20%He, and H270He30 stands for a mixture of ratio of 70%H2 to 30%He. The typical error bar for all phases is shown at upper right.

Appearance of H2O would result in a higher heat conduction capability than pure H2 and H2-He mixture, except in the range of 5.8–6.8 GPa. Furthermore, the predicted H2O liquid below the H2, He, and H2O liquid mixture layer inside the Uranus suggests that there should have heat sources flowing upwards within the Uranus (Arridge et al., 2014). Although we cannot quantify the heat propagation in those giant planets based on thermal conductivities of H2 and H2-He mixture obtained at room temperature, the low thermal conductivities of H2 and H2-He mixture in their liquid forms would still provide meaningful explanation for the poor heat conduction at the top layers of those giant planets.

Unlike terrestrial planets, there is no clear layer boundary within the gas/ice giant planets and, therefore, it is difficult to determine the radius of these giant planets at 1 bar condition. Furthermore, detailed temperature profiles, especially for the top layer, within giant planets are not well constrained. As mentioned above, the temperature at 1 bar was reported in the range of 72–165 K on the gas/ice giant planets, and temperature at 10 GPa was predicted at about 3000–4000 K for gas giant planets and about 1770–2000 K for ice giant planets (Guillot, 2005; Nettelmann et al., 2013). The thermal conductivity of liquid H2-He mixture is in the range of 0.70–1.0 W m−1 K−1 which is six to seven times lower than those of solid H2-He mixtures at low pressures. The low thermal conductivity of liquid H2-He mixture could result in a low heat loss, compared to those of solid H2-He mixture, and serve as a blanket-like effect. On such a basis, the low temperature existing on the surfaces of giant planets could be reasonably explained and also used to understand why those giant planets have cold surfaces but hot interiors.

Acknowledgments

This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN/06818-2019) to SRS, and by Academia Sinica and National Science and Technology Council of Taiwan, Republic of China, under Contract AS-IA-111-M02 and 110-2628-M-001-001-MY3 to WPH. WPH also acknowledges the fellowship from the Foundation for the Advancement of Outstanding Scholarship, Taiwan.

    Data Availability Statement

    Data used for the figures are available online (https://doi.org/10.7910/DVN/VL4LAQ).