If successfully developed, calorimetry at tens of GPa of pressure could help characterize phase transitions in materials such as high-pressure minerals, metals, and molecular solids. Here, we extend alternating-current calorimetry to 9 GPa and 300 K in a diamond anvil cell and use it to study phase transitions in H2O. In particular, water is loaded into the sample chambers of diamond-cells, along with thin metal heaters (1 μm-thick platinum or 20 nm-thick gold on a glass substrate) that drive high-frequency temperature oscillations (20 Hz to 600 kHz; 1 to 10 K). The heaters also act as thermometers via the third-harmonic technique, yielding calorimetric data on (1) heat conduction to the diamonds and (2) heat transport into substrate and sample. Using this method during temperature cycles from 300 to 200 K, we document melting, freezing, and proton ordering and disordering transitions of H2O at 0 to 9 GPa, and characterize changes in thermal conductivity and heat capacity across these transitions. The technique and analysis pave the way for calorimetry experiments on any non-metal at pressures up to ∼100 GPa, provided a thin layer (several μm-thick) of thermal insulation supports a metallic thin-film (tens of nm thick) Joule-heater attached to low contact resistance leads inside the sample chamber of a diamond-cell.

High pressures can cause insulators to become metals,1 metals to become insulators,2 normal metals to become superconductors,3 solids to melt,4 crystals to become amorphous,5 molecules to dissociate,6 or noble gases to chemically react.7 Detailed knowledge of such transformation is useful for understanding the Earth, other planets, and the chemistry and physics of materials, in general. But laboratory measurements of material properties across phase transformations are limited to the suite of probes compatible with high-pressure devices.

Calorimetry is one type of measurement missing from the suite of routinely used probes for diamond anvil cells, the static pressure devices capable of generating the highest pressures in laboratories. If it were successfully developed, diamond-cell calorimetry could be used to characterize thermal conductivity, heat capacity, and entropy across all of the types of transitions listed earlier. Calorimetry experiments would also improve the possibilities for discovering new phase transitions at high-pressure, such as liquid-liquid or liquid-amorphous transitions, which involve subtle structural changes but potentially large changes in entropy. Finally, if developed quantitatively, diamond-cell calorimetry could be used to determine Debye temperatures and test the Debye model at high-pressures.

Unfortunately, calorimetry has rarely been combined with diamond anvil cells. Pioneering work by a few groups has shown that it is possible to detect second order phase transitions at low temperatures and high pressures in diamond-cells using frequencies up to 10 kHz.8,9 But to make measurements at even higher pressures and/or at higher temperatures where thermal diffusivity is larger, we have previously shown that higher frequencies are required and that they are feasible using existing technology.10,11 Higher frequency techniques may also enable new measurements at ambient pressure, such as calorimetry of μm-thickness samples (e.g., Ref. 12) and specific heat spectroscopy of viscous liquids (e.g., Refs. 1314).

The most promising experimental result in Ref. 11 is that thin metal heaters can be used to heat and measure temperature oscillations in electrically insulating samples pressed between the tips of diamond anvils. Here, we extend AC calorimetry to 300 K at 9 GPa by adapting the third harmonic temperature measurement to high pressure for the first time.

We note that several groups have adapted a related technique that uses laser-heating instead of Joule-heating, time-domain thermoreflectance, to the diamond-cell and succeeded in several studies of transport measurements,15–18 but have yet to show that high-pressure data can constrain both thermal conductivity and heat capacity.

Our experimental method involves three basic steps, described in detail: (1) connecting a microscopic heater (μm-lengthscale) in a high-pressure environment to macroscopic electrical leads at ambient pressure, (2) connecting the leads to the electrical test equipment via a bridge circuit, and (3) driving AC current through the heater and measuring the resulting voltage oscillations. The data analysis procedure then uses the measured voltage oscillations to infer thermal properties.

Pressure and temperature are varied via established methods. In particular, we tighten and loosen screws to raise and lower pressure of the diamond cell, which we monitor with the ruby pressure scale.19 We vary background temperature using a liquid-nitrogen-cooled cryostat with a heated base, and monitor temperature on the diamond cell body with a silicon diode (LakeShore DT-670).

The results of three diamond-cell loadings with three different heaters are presented here. Heaters #1, #2 and #3 are summarized in Table I. Heaters #1 and #2 are gold thin-film heaters (20 nm-thick) sputtered onto a silicate glass substrate (product number AU.0100.CSR8 from Amsbio) and connected to two to four strips of platinum or gold foil by microsoldering (Fig. 1). Before soldering, the substrate is polished to 10 ± 2 μm thick, and ∼100 μm wide pieces are broken by pressing with a razor blade until an appropriately shaped piece is found. The platinum or gold foil is 2 ± 1 μm thick, and cut into ∼20 × 200 μm rectangles with a razor blade. The soldering is performed using a 12-Watt soldering iron (WM120) equipped with a hand-polished aluminum soldering tip (∼20 μm tip size) connected to one arm of a micromanipulator (Fig. 2) while the other arm holds a tungsten needle used to maintain the Pb-Sn solder in the correct position during soldering. The metal foil and glass substrate are stuck to the anvil culet with an adhesive (Crystalbond) in order to restrict their movement during soldering. Heater #3 is a razor-cut piece of 2 μm-thick platinum foil. In our lab, creating a heater from a platinum foil is routine, while soldering to the thin-film of gold results in unpredictable levels of contact resistance, from < a few Ωs (as in heaters #1 and #2) to tens of kΩ (in which case no data was collected). In the future, a more reliable method to create low contact resistance connections to thermally insulated thin-films inside diamond cell sample chambers would enable more efficient experimentation.

FIG. 1.

(a) Photo of the sample chamber at 8 GPa, 300 K. Four gold electrical leads are soldered onto the gold thin-film coating of a silicate glass substrate and compressed in a H2O medium. The thin-film heater is the exposed portion of gold between lumps of solder. (b) Schematic cross-section of the sample chamber (not to scale; ∼3-fold vertical exaggeration). Gold electrical leads (thicker yellow) soldered to a gold thin film (thin yellow), atop a glass substrate (white) in a H2O medium (blue) inside the hole of a cBN-epoxy gasket (dark grey) squeezed between the diamond anvils (light grey). Flakes of ruby (red) are embedded in the glue (light purple) that borders the glass substrate on two sides.

FIG. 1.

(a) Photo of the sample chamber at 8 GPa, 300 K. Four gold electrical leads are soldered onto the gold thin-film coating of a silicate glass substrate and compressed in a H2O medium. The thin-film heater is the exposed portion of gold between lumps of solder. (b) Schematic cross-section of the sample chamber (not to scale; ∼3-fold vertical exaggeration). Gold electrical leads (thicker yellow) soldered to a gold thin film (thin yellow), atop a glass substrate (white) in a H2O medium (blue) inside the hole of a cBN-epoxy gasket (dark grey) squeezed between the diamond anvils (light grey). Flakes of ruby (red) are embedded in the glue (light purple) that borders the glass substrate on two sides.

Close modal
FIG. 2.

Final steps in preparing for experiments with heater #2. (a) Soldering iron attached to one arm of a micromanipulator, in position to solder at the diamond tip. (b) Two platinum strips soldered to the gold thin film. (c) Heater #2 loaded inside the sample chamber of a diamond-cell with H2O medium at 1 GPa.

FIG. 2.

Final steps in preparing for experiments with heater #2. (a) Soldering iron attached to one arm of a micromanipulator, in position to solder at the diamond tip. (b) Two platinum strips soldered to the gold thin film. (c) Heater #2 loaded inside the sample chamber of a diamond-cell with H2O medium at 1 GPa.

Close modal
TABLE I.

The three heaters used in this study.

Heater #1Heater #2Heater #3
Length (μm) 50 ± 20 115 ± 20 250 ± 50 
Width (μm) 60 ± 20 55 ± 10 20 ± 5 
Thickness (μm) 0.02 ± 0.005 0.02 ± 0.005 2 ± 1 
Two-point resistance (Ω) 26 0.8 
Starting pressurea (GPa) 0.6 1.1 
Heater #1Heater #2Heater #3
Length (μm) 50 ± 20 115 ± 20 250 ± 50 
Width (μm) 60 ± 20 55 ± 10 20 ± 5 
Thickness (μm) 0.02 ± 0.005 0.02 ± 0.005 2 ± 1 
Two-point resistance (Ω) 26 0.8 
Starting pressurea (GPa) 0.6 1.1 
a

Pressures quoted are the lowest pressure at which the heater was used. Resistances quoted are at this pressure and 300 K.

To connect these heaters to macroscopic electrical leads at ambient pressure, the platinum or gold strips are pressed underneath a gasket into larger gold strips (∼3000 × 100× 10 μm) that are soldered to copper pins or foils that are glued to the cell body or to the gasket. The inner gasket is made of cubic boron nitride (cBN) mixed with epoxy (Epo-Tek 353 ND) and pressed inside a steel collar in the case of heaters #1 and #3. For heater #2, we follow the gasket design of Ref. 20: 50 nm Al2O3 is mixed with a UV curing epoxy, spread onto an indented and drilled brass gasket, cured, indented, and drilled with a near IR pulsed laser.

Three to four flakes of ruby are embedded in the adhesive that surrounds the substrate, a drop of water is placed on the gasket, and the cell is closed, completing the electrical circuit, trapping the water, and generating pressure as the gasket deforms inward. We note that the gasket crumbles slightly at the inner edge of the hole, sending some ceramic-epoxy mixture into the sample chamber. This crumbling led to the total collapse of gasket holes in the case of cBN gaskets pre-indented to ∼10 GPa between 1 mm-diameter diamond culets, likely because of a positive feedback between deformation of the cBN and hydration-induced weakening of uncompacted cBN. The successful cBN gaskets, by contrast, are indented to ∼20 GPa between 0.5 mm-diameter diamond culets, causing sufficient compaction of the cBN-epoxy mixture to enable trapping of water. Nonetheless, heater #2 is successfully used in a gasket pre-indented to ∼10 GPa. But in this case, the 50 nm Al2O3-based insulation is transparent, suggesting it has been well-compacted, and indeed the gasket traps water.

Ideally, this experiment would use four-point probe geometries to enable accurate monitoring of heater resistance. In practice, we use two, three, and four leads for heaters #3, #1, and #2. In particular, heater #3 uses a single strip of platinum that connects to four copper leads (two leads for each end of the platinum) outside the diamond cell. Heater #1 is connected to four strips of gold whose ends are soldered onto the thin-film heater. One of the four connections failed, so we rely on a three-point probe to estimate heater resistance without contribution from contacts or leads. Heater #2 uses two strips of platinum whose middle is soldered to the thin-film, creating four probes within the diamond cell, but the platinum-solder-gold connection may contribute contact resistance that is not detected in the four point probe measurement (see Fig. 3).

FIG. 3.

Schematic of the electrical setup, including a photo of the sample chamber seen through the cryostat window. Resistor values used here are R1 = 100 Ω, R2 = 47 Ω, and R3 = 1 kΩ.

FIG. 3.

Schematic of the electrical setup, including a photo of the sample chamber seen through the cryostat window. Resistor values used here are R1 = 100 Ω, R2 = 47 Ω, and R3 = 1 kΩ.

Close modal

The leads emerging from the diamond-cell are connected to a 3 × 3 cm electrical board, which is connected via coaxial cables to the electrical test equipment, as shown schematically in Fig. 3. The board and electronics are described briefly in Ref. 11, where they are used for test measurements on metal foils at ambient pressure and temperature.

The electronics board is centered around the bridge circuit shown in Fig. 3 and is prepared by (1) etching away the copper from 1/16 in.-thick copper clad board using nail-polish resist and ferric chloride etchant, (2) drilling holes for coaxial connectors and a bolt that secures the board within the cryostat, and (3) soldering the two capacitors, four resistors, and one instrumentation amplifier (AD8421) shown in Fig. 3, along with the coaxial connectors. The copper wire connects four points on the board to the four probes of the diamond cell circuit (or to each other, in the case of two- and three-point probes), labeled blue in the schematic. Approximately 0.5 m-long coaxial cables connect the board to feedthroughs at the top of the cryostat, which connect to the electrical test equipment via meter-long coaxial cables. The coaxial cables are likely necessary for maintaining a pure sinusoidal driving voltage through the 1.5 m of distance between the waveform generator and the electrical board, and perhaps also useful for delivering an undistorted signal to the oscilloscope, though neither requirement has been tested.

The electronics test equipment consists of (1) a two-channel 160 MHz, 14-bit digital waveform generator (BK4065) that delivers a sinusoidal driving and compensation voltages, (2) a differential preamplifier that differences the voltages from the two sides of the four-point probe (SRS 560), (3) a DC power supply that powers the in-amp (Mastech HY3005), and (4) a two-channel 500 MHz, 8-bit digital oscilloscope (LT342) that measures the signal from the output of the in-amp in one channel, and the output from the preamplifier in the other. In Ref. 11 we showed that upgrading from an oscilloscope to a lock-in amplifier does not improve the ratio of signal to background since the background is spurious harmonics generated in our circuit (likely within the electrical test equipment).

After loading the high-pressure cell and making all electrical connections, we then begin electrical measurements. A ±1 V, 1 kHz sine-wave driving voltage is delivered from channel 1 of the waveform generator, through the 47 Ω reference resistor, through the high-pressure heater, to the ground. The heater resistance is measured on channel 2 of the oscilloscope, using the pseudo-four-point probe described above. The gain of the in-amp is measured on channel 1 of the oscilloscope. To test the bandwidth of our electronics, we vary the frequency, and find that up to 1 MHz, gain is nearly constant (i.e., variations are much smaller than the uncertainties in our measurements).

Then we proceed to the calorimetry measurement by increasing the driving voltage to a suitably high value (e.g., the ±5 V at 1 kHz shown in Fig. 4). This fundamental-mode voltage oscillation generates a second harmonic oscillation in electrical power, which causes an oscillation in heater temperature and an associated oscillation in resistance, which generates a third harmonic oscillation in voltage. To measure this “3ω” voltage, we adapt the design of Ref. 13, as our previous work.11 In particular, we drive the dummy arm of the bridge circuit via channel 2 of the waveform generator, and use the in-amp to difference the voltage created at the center of the dummy arm from the voltage at the center of the sample arm. The resulting voltage, multiplied by gain, is the output from the in-amp and digitized by the oscilloscope. First, we tune the voltage of channel 2 of the waveform generator so that oscillations at channel 1 of the oscilloscope are minimized. After tuning by 20% in voltage, then 10%, then 5%, etc., we arrive at a final compensation voltage, Vc, with a precision of less than 1%. This 1 kHz tuning procedure is applied once at each pressure, temperature, and amplitude of the driving voltage.

FIG. 4.

Example of raw data at 8 GPa, 272 K, 1 kHz, and ±5 V, shown in the time domain (top) and frequency domain (bottom). The nominal driving voltage from the waveform generator is shown in black, while the voltage measured by the oscilloscope is shown in blue.

FIG. 4.

Example of raw data at 8 GPa, 272 K, 1 kHz, and ±5 V, shown in the time domain (top) and frequency domain (bottom). The nominal driving voltage from the waveform generator is shown in black, while the voltage measured by the oscilloscope is shown in blue.

Close modal

Then we scan the driving frequency, f, from 10 Hz to 300 kHz, and measure the amplitude of third harmonic voltage oscillation at each frequency by collecting 100 cycles with 100 data points per cycle, uploading the voltage time-series to a PC, and calculating the Fourier component at the third harmonic frequency.

In our experience, measurement precision does not improve by tuning phase or by retuning amplitude at frequencies up to 100 kHz. From 100 kHz to the maximum frequency used here, 300 kHz, retuning of phase and amplitude would reduce the first harmonic amplitude, but in our measurements, there was no need to do so; measured voltages did not exceed 1 V, well within the range of both in-amp and oscilloscope.

Our analysis requires the six measurements described above. They are pressure, P, background temperature, Tbknd, and four electrical measurements: four-point-probe resistance, Rsam4pt, gain, G, compensation voltage, Vc, and amplitude of third harmonic voltage, V3ω. We impose the amplitude of driving voltage, Vd, and frequency, f.

First, we use values Vc, Vd and the resistances of two dummy and one reference resistor, R1, R2, and R3, to calculate the two-point resistance across the sample plus leads

Rsam2pt=VcR3(R0+R2)(R0+R1+R3)(VdVcR3R0+R1+R3),
(1)

where R0 = 50 Ω is the internal resistance of the waveform generator. Equation (1) is derived from the equality of fundamental mode voltage at midpoints of each arm of the bridge circuit, which assumes Ohm's law and that the bridge is well-tuned in our experiments.

We then calculate the amplitude of temperature oscillation, T2ω, defined as the Fourier component of temperature at the 2nd harmonic frequency (i.e., absolute temperature equals Tbknd+T2ω(1+sin(2ωt+ϕ)) plus negligibly small oscillations at other frequencies). We use Eq. (1) of Ref. 11 

T2ω=2(R0+R2+Rsam2pt)2V3ωαVdRsam4pt(R0+R2)G,
(2)

where α=1RdRdT is the temperature coefficient of resistance inferred from measurements Rsam4pt and T during the cooling and heating cycles. Eq. (2) is derived in Ref. 11, and is similar to Eqs. (11) and (12) of Refs. 13 and 21. In all cases, the main assumptions are that third-harmonic voltage oscillations are generated solely in the heater and are due to uniform temperature-oscillations of the heater. Here, we plot the amplitudes of temperature oscillations inferred from third-harmonic voltages in the second panel of Fig. 5.

FIG. 5.

Example of data processing at a single pressure and temperature (8 GPa, 272 K), and a range of frequencies and driving amplitudes (10 Hz to 300 kHz; ±5 V in blue, ±4 V in green). The solid black curve in the third panel is the fit to the complete data set, with the dashed black curve representing the two components that sum to the fit. The bottom two panels show fitted values of KBC and E for data truncated at maximum frequencies from 2 kHz to 300 kHz.

FIG. 5.

Example of data processing at a single pressure and temperature (8 GPa, 272 K), and a range of frequencies and driving amplitudes (10 Hz to 300 kHz; ±5 V in blue, ±4 V in green). The solid black curve in the third panel is the fit to the complete data set, with the dashed black curve representing the two components that sum to the fit. The bottom two panels show fitted values of KBC and E for data truncated at maximum frequencies from 2 kHz to 300 kHz.

Close modal

We then calculate the resistance of the system to changes in temperature, which we call “effective thermal conductance” and denote as Kth. It accounts for both the heat capacity of the metal heater (which is negligible for thin-film heaters #1 and #2) and conductance to the surroundings. It is defined by

Kth=p2ωT2ω,
(3)

where p2ω=12Iω2Rsam4pt for Iω=Vd/(R0+R2+Rsam2pt).

We use the error analysis of Ref. 11 to estimate uncertainties in V3ω, T2ω, and Kth. In Ref. 11, we showed that our electrical test equipment generates ∼1 part in 104 third harmonic distortion in voltage (i.e., –80 dBc), as evidenced by testing of metal foil standards at ambient conditions and of dummy samples (i.e., off-the-shelf resistors). We assume that this is the dominant source of uncertainty in V3ω, T2ω, and Kth.

We then fit the effective thermal conductances at all frequencies and voltages to the same two-parameter model used in Ref. 11. This model is motivated by one-dimensional models of heat transfer, which are only realistic in the high-frequency limit when the thermal penetration depth, K2fρC, is much less than the ∼50 μm heater width. At our highest heating frequencies (2f ∼ 600 kHz), we estimate the thermal penetration depths of ∼10 μm (K = 3 W/m/K, ρ = 103 kg/m3, C = 4200 J/kg/K). Therefore, we only discuss relative changes in the fitted parameters, not their absolute values. Our interpretations also assume that the sample has time to relax during all temperature cycles imposed here; i.e., we are probing relaxed values of enthalpy fluctuation.

Effective thermal conductance is modeled as the sum of two terms with different frequency-dependences, Kth=KBC+E2ω. Each fitting term and their sum (i.e., the total fit) are shown in the third panel of Fig. 5. Fits and their uncertainties are calculated using the “curve_fit” function within the scipy.optimize library of python, weighted via uncertainties in Kth. We also check sensitivity to the assumed bandwidth by plotting fitted values of KBC and E versus the maximum frequency used in the fit (e.g., bottom two panels of Fig. 5). In the data from heaters #1 and #2, fits are not sensitive to the bandwidth provided it is at least ∼30 kHz, so we therefore assume uncertainties in KBC and E calculated from our full frequency scan, 10 Hz to 300 kHz. The bandwidth of third harmonics generated in heater #3 is more limited due to its low impedance (∼0.8 Ω), so we fit from 10 Hz to 5 kHz instead of 300 kHz.

We refer to the first term, KBC, as the “boundary thermal conductance” since it reflects the rate of heat flow to a fixed-temperature boundary condition in a one-dimensional model. In the limit of an extremely thin but wide substrate and sample, and of infinite thermal conductivity diamonds, this heat flow to the boundary reflects material properties and thicknesses of substrate and sample: KBC = Ksubdsub + Ksamdsam for conductivity K, thickness d, and subscripts “sub” and “sam” referring to the substrate and sample.

We refer to the second term, E, as “bulk effusivity,” since it describes transport of heat into the sample and substrate. Thermal effusivity is a material property defined as the geometric mean of thermal conductivity and volumetric heat capacity (i.e., effusivity = ρCK where ρ, C, and K are density, specific heat per mass, and thermal conductivity). It describes the ability of a material to transport to or from its surface. When the frequency is large enough so that thermal diffusion is limited to lengthscales smaller than the heater width and substrate thickness, our fitting parameter E can be understood in terms of effusivity: E(Ksubρsubcsub+Ksamρsamcsam)A, where ρ and c are density and specific heat capacity, and A is the surface area of the heater. A third term should be used when the lengthscale of thermal diffusion is less than the thickness of the heater, i.e., for heater #3, but we omit it here, opting instead to interpret the fitted values of KBC and E qualitatively when using heater #3. We also note that the measurements presented here were not accurate enough to allow inference of the absolute value of the sum of effusivities of the sample and substrate, (Ksubρsubcsub+Ksamρsamcsam), but rather the sign and approximate magnitude of the changes of this quantity across phase temperature-induced phase transitions. The most insidious sources of error are likely contact resistance, which causes spurious harmonics, and inhomogeneity in the heater width, which causes inhomogeneous heat deposition. Neither source of error seems to cause large changes in the ratio of E or KBC across the phase transitions studied as evidenced by the reproducibility of relative changes in E and KBC.

The difference between specific heat at constant pressure and constant volume, cp versus cv, is likely several percent at the temperatures studied here (cp/cv = 1 + αγT for thermal expansion α, Gruneisen's parameter γ, and temperature T), which is much smaller than the changes detected in our experiments. We therefore defer detailed discussion of the mechanical boundary condition in a rapidly heated diamond cell for future studies, but simply note that at high-enough frequency, thermal diffusion is limited to a small fraction of the sample chamber, ensuring that a liquid sample can be studied at a nearly constant pressure. The boundary condition for a solid sample, however, depends on the degree of mechanical relaxation achieved during thermal oscillations, which is a complex function of sample and gasket geometry, material properties, and deformation history. Detailed modeling for the case of a continuously heated diamond-cell can be found in Ref. 22.

Heaters #1, #2, and #3 were used to measure the boundary thermal conductance and bulk effusivity during temperature scans from 300 K to ∼260 K, or in some cases, to 200 K, at pressures of 6 to 9 GPa, 0.6 GPa, and 1.1 GPa, respectively. In all cases, distinct steps in boundary thermal conductance are seen near temperatures corresponding to phase transitions identified in the published literature (Figs. 6–11).23 Hence, we identify phase transition temperatures by the midpoint of the rapid change in boundary thermal conductance. Since melting almost always takes place at the thermodynamic melting point,24 we assume that the large hysteresis detected upon cycling temperature at 0.6 and 1.1 GPa is due to supercooling. In particular, Fig. 8 shows 28 K of supercooling, while Fig. 7 shows at least 30 K of supercooling (melting is not obvious in the data, possibly because the thermodynamic melting point is above 300 K at the pressure of the experiment, 1.1 ± 0.3 GPa). In the case of heaters #1 and #2 (i.e., the thin-film heaters), we detect jumps in bulk effusivity at the same temperatures as jumps in boundary thermal conductance. Hence, our measurements show that the technique described above is compatible with diamond-cell experiments from 200 to 300 K and 0 to 9 GPa. Moreover, the details of the changes in boundary thermal conductance and bulk effusivity reveal new information about phase transitions among several of the high-pressure phases of H2O.

FIG. 6.

The ice VII-VIII phase transition detected via calorimetry at 8.1 ± 0.3 GPa. Heater resistance, R, and the two fit parameters in our thermal model, KBC and E, are plotted as a function of temperature during cooling (blue) and heating (red). Phase transitions are detected at 275 ± 2 K upon cooling and at 278 ± 4 K upon heating, using scan rates of 4 K/h and 3 K/h, respectively. The pressure drifts from 7.9 to 8.5 GPa during cooling and reverses during heating, but the range of pressures during the phase transitions is more limited: from 8.0 to 8.1 GPa, with 0.2 GPa uncertainty in each pressure measurement.

FIG. 6.

The ice VII-VIII phase transition detected via calorimetry at 8.1 ± 0.3 GPa. Heater resistance, R, and the two fit parameters in our thermal model, KBC and E, are plotted as a function of temperature during cooling (blue) and heating (red). Phase transitions are detected at 275 ± 2 K upon cooling and at 278 ± 4 K upon heating, using scan rates of 4 K/h and 3 K/h, respectively. The pressure drifts from 7.9 to 8.5 GPa during cooling and reverses during heating, but the range of pressures during the phase transitions is more limited: from 8.0 to 8.1 GPa, with 0.2 GPa uncertainty in each pressure measurement.

Close modal
FIG. 7.

Freezing of water ice at 1.1 GPa using heater #2. Heater resistance, R, and the two fit parameters of our thermal model, KBC and E, are plotted versus temperature during cooling (blue) and heating (red). Freezing is clearly seen at 249 ± 2 K, whereas melting is unclear, though the thermal and electrical properties return to their approximate starting values. The increase in temperature after freezing reflects the decrease in thermal conductance to the diamonds, which overwhelms the steady decrease in the background temperature, but which does not cause reversal of the freezing transition.

FIG. 7.

Freezing of water ice at 1.1 GPa using heater #2. Heater resistance, R, and the two fit parameters of our thermal model, KBC and E, are plotted versus temperature during cooling (blue) and heating (red). Freezing is clearly seen at 249 ± 2 K, whereas melting is unclear, though the thermal and electrical properties return to their approximate starting values. The increase in temperature after freezing reflects the decrease in thermal conductance to the diamonds, which overwhelms the steady decrease in the background temperature, but which does not cause reversal of the freezing transition.

Close modal
FIG. 8.

Melting and freezing of water ice at 0.6 GPa using heater #3 during two cycles of temperature between 300 and ∼220 K. Heater resistance, R, and the two fit parameters of our thermal model, KBC and E, are plotted versus temperature during cooling (blue) and heating (red). All transitions in KBC are sharp: freezing is detected over a 0.5 K temperature interval at 219 ± 1 K, while melting is detected over a 2 K temperature interval at 247 ± 1 K. The fit value of bulk effusivity, E, does not show melting and freezing, likely because the low impedance of heater #3 limits third harmonic signals to small values at the high frequencies needed to constrain it.

FIG. 8.

Melting and freezing of water ice at 0.6 GPa using heater #3 during two cycles of temperature between 300 and ∼220 K. Heater resistance, R, and the two fit parameters of our thermal model, KBC and E, are plotted versus temperature during cooling (blue) and heating (red). All transitions in KBC are sharp: freezing is detected over a 0.5 K temperature interval at 219 ± 1 K, while melting is detected over a 2 K temperature interval at 247 ± 1 K. The fit value of bulk effusivity, E, does not show melting and freezing, likely because the low impedance of heater #3 limits third harmonic signals to small values at the high frequencies needed to constrain it.

Close modal
FIG. 9.

Phase transitions detected in this study during cooling (blue) and heating (red), and boundaries inferred previously (black curves, with dashed lines implying extrapolation).23 

FIG. 9.

Phase transitions detected in this study during cooling (blue) and heating (red), and boundaries inferred previously (black curves, with dashed lines implying extrapolation).23 

Close modal
FIG. 10.

Temperature scans of the cell containing heater #3, at 5 to 6 GPa. Pressures at the transitions are 5.8, 5.7, 5.4, and 5.3 GPa, from the leftmost to the rightmost column, with typical uncertainty of 0.3 GPa.

FIG. 10.

Temperature scans of the cell containing heater #3, at 5 to 6 GPa. Pressures at the transitions are 5.8, 5.7, 5.4, and 5.3 GPa, from the leftmost to the rightmost column, with typical uncertainty of 0.3 GPa.

Close modal
FIG. 11.

Temperature scans of the cell containing heater #3, at 7 to 8 GPa. Pressures at the transitions are 6.8, ∼8, 8.8, and 8.7 GPa, from the leftmost to the rightmost column, with typical uncertainties of 0.3 GPa. Pressure was increased during the cooling run plotted in the second column. The rightmost column corresponds to slow heating and cooling rates (–3 and +4 K/h), whereas the other temperature scans in this figure and in Fig. 10 correspond to fast cooling rates (∼10 K/h).

FIG. 11.

Temperature scans of the cell containing heater #3, at 7 to 8 GPa. Pressures at the transitions are 6.8, ∼8, 8.8, and 8.7 GPa, from the leftmost to the rightmost column, with typical uncertainties of 0.3 GPa. Pressure was increased during the cooling run plotted in the second column. The rightmost column corresponds to slow heating and cooling rates (–3 and +4 K/h), whereas the other temperature scans in this figure and in Fig. 10 correspond to fast cooling rates (∼10 K/h).

Close modal

Pressure was measured during cooling and heating of the cell with heater #1 at 6 to 9 GPa. It typically drifted upwards by ∼15% during cooling and down by ∼15% during heating, which is expected for the Mao-Bell style of diamond-cell used with heater #1. Pressure was measured only at ambient temperature when using the other cells, but since they used a cell design that typically exhibit less pressure drift, a ∼5 cm version of the cell of Ref. 25, we assume pressure drifted by at most 30%. Hence, we plot transition pressures that equal the value measured at ambient temperature and with ±30% error bars for data from heaters #2 and #3 (Fig. 9, with 30% or 0.2 to 0.3 GPa horizontal error bars for heaters #1 and #2 at 0.6 to 1.1 GPa). The pressures plotted at 6 to 9 GPa correspond to pressures measured during the cooling and heating cycles with heater #1, so the error bars represent pressure drift during transitions to and from ice VIII.

Heater #1 shows transitions at 6 to 9 GPa between ice VII and ice VIII. Transitions from VII to VIII are characterized by a 40% to 50% increase in boundary thermal conductance and a decrease in bulk effusivity of a few tens of percent, spread over a ∼4 K temperature range (Figs. 6, 10, and 11). The transition is reversible, with hysteresis of ∼3 K in the slow cooling and heating example (3 to 4 K/h) shown in Fig. 6. The transition was reversed in all nine temperature cycles monitored for this study, at pressures between 6 and 9 GPa, with varying cooling and heating rates. Heater resistance during temperature cycling is also presented in Figs. 10 and 11, which shows small, gradual, and nearly reversible changes of ∼3% as temperature cycles between 300 and 260 K, confirming that the transition detected in boundary thermal conductance and bulk effusivity derives from the heater's surroundings rather than the heater itself.

Figure 9 shows that the transitions between ice VII and VIII match the phase boundary inferred previously. Here we also constrain the kinetics of the phase transition (Fig. 12) by comparing the temperature of the transition during the four temperature cycles performed at 8.2 ± 0.6 GPa. Whereas the temperatures error bars in our equilibrium phase diagram, Fig. 9, reflect the widths of transitions, the ±1 K error bars in the plot of kinetics, Fig. 12, reflect the uncertainty is identifying the midpoint of the transition.

FIG. 12.

Kinetic dependence of ice VII-VIII transition temperature at 8.2 ± 0.6 GPa. The temperature at the midpoint of the transition in KBC is plotted versus heating or cooling rate (red or blue, respectively).

FIG. 12.

Kinetic dependence of ice VII-VIII transition temperature at 8.2 ± 0.6 GPa. The temperature at the midpoint of the transition in KBC is plotted versus heating or cooling rate (red or blue, respectively).

Close modal

Figure 12 shows a subtle trend in the temperature of transition during heating: the temperature of proton-disordering increases by ∼3% (or 6 K) as the heating rate increases from 3 to 10 K/h. This small effect cannot be due to temperature gradients in our cryostat, since the Si-diode thermometer was taped to the cell body, minimizing the temperature difference between the thermometer and sample. Specifically, in a footnote26 we estimate the temperature difference during the most rapid heating presented in Fig. 12, 10 K/h, to be at most 0.5 K.

Whereas the pressure-temperature conditions of the phase transition and the existence of a few K hysteresis were noted decades ago,27,28 the present constraint on kinetics was not: the four data points from heating experiments show that the temperature of proton-disordering increases with increasing heating rate (Fig. 12). Whether this effect is intrinsic to H2O is a question for future research; it is possible nucleation of the proton-disordered ice VII happens on a surface (e.g., Au-coated glass or diamond), and thus we are detecting kinetics of a specific variety of ice VII nucleation.

The magnitude of change in boundary thermal conductance and bulk effusivity reveals a large decrease in specific heat upon ordering of ice VII. Since the substrate contributes an unknown amount to total conductance and bulk effusivity, the step-function changes plotted here represent lower-bounds on the change in the properties of H2O. And since the changes are reversed in almost all temperature cycles, we expect that the geometry of heater and surroundings changes negligibly during the thermal cycles. Therefore, we conclude that thermal conductivity of H2O increases by at least 40% across the VII-VIII transition. The fitting parameter E is more difficult to interpret quantitatively than KBC, since our data only constrains E in the highest frequency range (∼10 kHz to 200 kHz). Nevertheless, the fitted value of E decrease substantially (a few tens of percent) across the ice VII-VIII transition in all our temperature runs (Figs. 6, 10, and 11), indicating that the intrinsic effusivity of H2O likely decreases by tens of percent, and that at least it does not increase. Since effusivity squared times thermal conductivity is volumetric specific heat, ρc, we conclude that specific heat decreases by at least 40% between ice VII and VIII. While we know of no previous publication reporting specific heat of ice VII and VIII, we can compare our result to a different proton ordering transition in H2O: calorimetry at 0.16 GPa shows that the proton-ordering of KOH-doped ice Ih to ice XI causes a ∼10% change in heat capacity.29 By comparison, our inference of >40% decrease in heat capacity suggests that the proton-ordering of undoped ice VII has a much larger effect on the accessible degrees of freedom. The ∼40% thermal conductivity change measured here is consistent with a past study that found that K increases from 3 to 6 W/mK from ice VII to VIII at 2.2 GPa.30 

Thermal conductivity of H2O is also seen to increase during freezing at 0.6 GPa, using heater #3 to monitor thermal properties. Figure 8 shows that the total conductance increases ∼2-fold when it freezes at 0.6 GPa, which is consistent with literature values of thermal conductivity of liquid H2O at ambient temperature and ice V at 0.6 GPa.30 

The thermal properties measured at 1.1 GPa using heater #2, on the other hand, do not match our expectation of freezing and melting; fitted values of boundary thermal conductance of H2O decrease by at least 30% upon cooling at 250 K and 1.1 GPa (Fig. 7). The step function change likely corresponds to freezing of a supercooled liquid since we expect no other large change in material properties, but the frozen sample is unlikely to be crystalline since a crystal's thermal conductivity is typically larger than that of the iso-chemical liquid. Indeed, previous measurements on crystalline H2O show that its thermal conductivity exceeds that of the liquid, regardless of pressure.30 In our case, the frozen H2O could be nano-crystalline or glassy.

Alternatively, the inferred decrease in conductance could have been due to something other than the material properties of H2O, such as a change in chemistry, geometry or electrical circuit. These possibilities seem unlikely. First, there is no evidence for an irreversible decrease in boundary thermal conductance or bulk effusivity, arguing against the possibility of a chemical reaction or change in the sample geometry causing the decrease in effective thermal conductance. In fact, upon warming to room temperature, the fitted value of boundary thermal conductance increases to a higher value than the starting value. There is a small (2%) decrease in heater resistance at 250 K upon cooling, which could be due to a decrease in contact resistance. We would expect this to cause a decrease in measured third harmonic and hence an increase in boundary thermal conductance. On the other hand, we cannot rule out the possibility that the freezing transition also caused an increase in contact resistance somewhere in the circuit in such a way that the total change in boundary thermal conductance is negative. Still, the simpler explanation is that we synthesized a nano-crystalline or glassy form of H2O at 1.1 GPa and 250 K.

High-pressure modulation calorimetry using a third-harmonic technique for temperature measurement and diamond cell for pressure generation has been demonstrated. It is shown to be a sensitive detector of phase transitions from 0 to 9 GPa of pressure. Moreover, changes in boundary thermal conductance and bulk thermal effusivity of the H2O sample studied here help constrain the thermal conductivity and heat capacity of H2O at high pressure.

As compared to previous techniques using modulated heating sources and thermocouples that are separated in space,8,31–33 the third-harmonic technique is better-suited to quantitative measurements of the ng-mass samples, typical of diamond-cell experiments. The reason is that by using a heater that is its own thermometer, the third-harmonic technique enables calorimetric measurements of materials solely within ∼10 μm of the heater, with the potential to reduce this lengthscale to ∼1 μm if higher-frequencies are employed.

As compared with the previously established technique of time-domain thermoreflectance,17,34 our Joule-heating technique has the advantage of allowing direct measurements of heat deposited, opening a new pathway to quantitative measurements of heat capacity with fewer fitting parameters than in the case of time-domain thermoreflectance. Still, further technique development is required to test the accuracy of our measurements, starting with the fabrication of thin-film heaters with a uniform width (see Ref. 11 for further discussion).

Two major laboratory challenges are involved: making circuits with low contact resistance on the tip of a diamond and maintaining low contact resistance at high-pressure. Gold and platinum foils soldered to thin-film gold heaters provide the highest bandwidth data presented here (∼300 kHz in driving voltage, 600 kHz in heating frequency), but improvements in micro-soldering or microwelding are needed to make this measurement routine. Such improvements could also enable absolute measurements of thermal conductivity and heat capacity over a wide range of pressures.

We thank Maddury Somayazulu and Paul Goldey for advice in experimental design, and Alexander Gavriliuk for providing diamond cells. Support was provided by the Carnegie Institution for Science, the U.S. DOE/BES (Award No. DE-FG02-99ER45775) and the U.S. DOE/NNSA (Award No. DE-NA-0002006, CDAC).

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