EAGER: Testing New Formulae for Pressure Derivatives of Specific Heat, Thermal Conductivity, and Thermal Diffusivity

Heat flow is ubiquitous. How efficiently matter transports heat is of fundamental importance. Understanding heat transport is critical to numerous engineering and scientific endeavors, such as designing tiny electronic devices or modeling the cooling rate of large planetary bodies.

The most accurate method for measuring heat transport properties (laser-flash analysis) has shown that values from diverse types of solids depend on the length over which heat is flowing. Static properties (e.g., density) do not behave in this manner, so tests against length were not made earlier. This finding led to new formulae describing thermal dynamic behavior (conductivity, diffusivity, and heat capacity) as functions of pressure.

The formulae describe steady-state conditions. This contrasts with equations of classical thermodynamics developed in the 1800s to describe idealizations such as constant temperature. These findings are critical for Geophysics because matter in Earth’s deep interior is under extreme pressures.

Here, the researchers will further test the formulation which they previously validated for a limited number of solids and over a limited range of temperature. They now quantify heat properties in a wider range of solids, and other states of matter, and over a greater temperature range. They use a new laser flash apparatus for liquids (water in particular); the new apparatus allows measuring heat capacity and attains low temperature.

They re-analyze data on gases available in the literature. The goal is extreme accuracy in the measurement, notably that of initial slope and temperatures near ambient conditions. Both the old and the new formulation for thermal properties are tested.

This project may initiate a paradigm shift in the way we quantify heat flow. Its outcomes improve our understanding of the microscopic mechanism responsible for it. They have potentially wide repercussions for pure and applied physical sciences.

The project also supports a female scientist with disability and provides training to undergraduate students at University of Washington.

Discovery that thermal diffusivity (D) and thermal conductivity (k) of insulators, semi-conductors, metals, alloys, and glasses depends on the length-scale (L) of measurements has repercussions for pure and applied physical sciences. Linear dependence on L for small length-scales agrees with dimensional analysis of Fourier’s heat equation. It shows that results from diamond anvil cell experiments are problematic, foremost because these are benchmarked against 1 atm data for L > 100 times larger.

Reliable data below 2 GPa pressure exist on mm-sized samples from well-worn methods. Analyzing these results for 25 solids of diverse bond type show that the logarithmic pressure derivative of specific heat (cP) equals -1 times the linear compressibility. Mathematical analysis allowing for k depending on L (i.e., volume) also related its logarithmic pressure response to equation-of-state properties, and likewise for D.

The new k vs. P formula was confirmed against reliable data for 20 solids. Therefore, the team’s preliminary tests suggest, but do not prove, that the new formulae are thermodynamic identities describing steady-state conditions, which is a commonly encountered restriction.

Isothermal is not, because thermal emissions are ubiquitous. Thus, the time-independent formulations from classical thermodynamics do not describe properly the time-depended conditions in the Earth where heat is flowing. To test the new formula for cP vs.

P, databases already in existence can be used. Since a generally applicable formula for steady state is the goal, diverse states of matter and bond type are being tested. The researchers also collect data on liquids and ices (e.g., water, metals melting near 298 K).

They use a new laser flash apparatus – the LFA467 instrument and its commercial low-pressure cell to 10 MPa - covering ~100 to 500°C, that simultaneously measures D and cP. Spurious radiative transfer is removed via coating the interior of the cell with graphite. A dilatometer spanning this T range is used when density vs T is not available, to constrain k.

Using machine learning to parameterize cP, D, and k vs. P, L, T, will either distinguishing whether the new or previous formulae are correct, or will lead to new and accurate formulae, permitting extrapolation to high pressure inside Earth. Thus, the proposed work provides basic physics that is essential to Geophysics.

Confirmed P derivatives of cP and k (or D) pertain to any process inside Earth and other large bodies. The work outcomes lead to improving accuracy of geophysical and petrologic models and has potential to improve our understanding of planetary interiors. Since new thermodynamic identities have not been developed for ~100-years, other physical sciences, such as study of chemical reactions, may be impacted.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.