# Modulation calorimetry in diamond anvil cells II: Joule-heating design and prototypes

## TL;DR

This paper demonstrates the feasibility of using high-frequency Joule heating in diamond anvil cells to measure heat capacity and thermal effusivity of materials under high pressure, with prototypes and numerical models showing promising accuracy.

## Contribution

It introduces a practical Joule-heating calorimetry method in diamond anvil cells, achieving accurate measurements despite experimental challenges, advancing high-pressure material characterization techniques.

## Key findings

- Numerical models show heat capacity can be measured at ~1 MHz with <10% accuracy.
- Prototypes measure specific heats of Fe, Pt, Ni with 20-30% accuracy.
- Thermal effusivity can be measured with ±6% accuracy using a thin-film heater.

## Abstract

Part I shows that quantitative measurements of heat capacity are theoretically possible inside diamond anvil cells via high-frequency Joule heating (100 kHz to 10 MHz), opening up the possibility of new methods to detect and characterize transformations at high-pressure such as the glass transitions, melting, magnetic orderings, or the onset of superconductivity. Here we test the possibility outlined in Part I, using prototypes and detailed numerical models. First, a coupled electrical-thermal numerical model shows that specific heat of metals inside diamond cells can be measured directly using $\sim 1$ MHz frequency, with $< 10%$ accuracy. Second, we test physical models of high-pressure experiments, i.e. diamond-cell mock-ups. Metal foils of 2 to 6 $\mu$m-thickness are clamped between glass insulation inside diamond anvil cells. Fitting data from 10 Hz to $\sim 30$ kHz, we infer the specific heat capacities of Fe, Pt and Ni with $\pm 20$ to $30%$ accuracy. The electrical test equipment generates -80 dBc spurious harmonics which overwhelm the thermally-induced harmonics at higher frequencies, disallowing the high precision expected from numerical models. An alternative Joule-heating calorimetry experiment, on the other hand, does allow absolute measurements with $< 10%$ accuracy, despite the -80 dBc spurious harmonics: the measurement of thermal effusivity, $\sqrt{\rho c k}$ ($\rho$, $c$ and $k$ being density, specific heat and thermal conductivity), of the insulation surrounding a thin-film heater. Using a $\sim 50$ nm-thick Pt heater surrounded by glass and 10 Hz to 300 kHz frequency, we measure thermal effusivity with $\pm 6%$ accuracy inside the sample chamber of a diamond anvil cell.