^{1,2,3}and Alexander F. Goncharov

^{1}

### Abstract

The time-dependent temperature distribution in the laser-heated diamond anvil cell (DAC) is examined using finite element simulations. Calculations are carried out for the practically important case of a surface-absorbing metallic plate (coupler) surrounded by a thermally insulating transparent medium. The time scales of the heat transfer in the DAC cavity are found to be typically on the order of tens of microseconds depending on the geometrical and thermochemical parameters of the constituent materials. The use of much shorter laser pulses (e.g., on the order of tens of nanoseconds) creates sharp radial temperature gradients, which result in a very intense and abrupt axial conductive heat transfer that exceeds the radiative heat transfer by several orders of magnitude in the practically usable temperature range (<12 000 K). In contrast, the use of laser pulses with several *μ*s duration provides sufficiently uniform spatial heating conditions suitable for studying the bulk sample. The effect of the latent heat of melting on the temperature distribution has been examined in the case of iron and hydrogen for both pulsed and continuous laser heating. The observed anomalies in temperature-laser power dependencies cannot be due to latent heat effects only. Finally, we examine the applicability of a modification to the plate geometry Ångström method for measurements of the thermal diffusivity in the DAC. The calculations show substantial effects of the thermochemical parameters of the insulating medium on the amplitude change and phase shift between the surface temperature variations of the front and back of the sample, which makes this method dependent on the precise knowledge of the properties of the medium.

We thank Z. Konopkova and P. Lazor for help in setting up FE calculations, V. V. Struzhkin for useful discussions, S. Gramsch, R. S. McWilliams, and D. A. Dalton for important comments and suggestions on the manuscript. We acknowledge support from NSF EAR 0711358 and EAR-1015239, Carnegie Institution of Washington, and DOE/NNSA (CDAC).

I. INTRODUCTION

II. COMPUTATIONAL METHOD

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Laser heating
- 28.0
- Diamond anvil cells
- 21.0
- Thermal conductivity
- 21.0
- Finite element methods
- 12.0
- Latent heat
- 11.0

## Figures

Model of the DAC assembly used in the FE calculations.

Model of the DAC assembly used in the FE calculations.

Finite-element calculations of the temperature profiles in the axial direction of the DAC cavity in the center of a laser-heating spot for the case of the symmetric double-sided heating. (a) Comparison of the temperature profiles corresponding to the time when the maximum of temperature is reached at the coupler–medium interface for continuous, 2 *μ*s pulsed, and 10 ns pulsed laser heating regimes. (b) Evolution of the temperature profile with time for 10 ns pulsed laser heating; the maximum of the laser intensity (shown in (a)) corresponds to the 20 ns time. Thin vertical lines correspond to the coupler-medium interface. The medium–diamond interfaces correspond to ±8 *μ*m abscisses.

Finite-element calculations of the temperature profiles in the axial direction of the DAC cavity in the center of a laser-heating spot for the case of the symmetric double-sided heating. (a) Comparison of the temperature profiles corresponding to the time when the maximum of temperature is reached at the coupler–medium interface for continuous, 2 *μ*s pulsed, and 10 ns pulsed laser heating regimes. (b) Evolution of the temperature profile with time for 10 ns pulsed laser heating; the maximum of the laser intensity (shown in (a)) corresponds to the 20 ns time. Thin vertical lines correspond to the coupler-medium interface. The medium–diamond interfaces correspond to ±8 *μ*m abscisses.

Finite element simulation of the coupler surface temperature history. (a)–(c) are corresponding to 2 *μ*s, 10 ns, CW laser heating, respectively. The dashed curve corresponds to calculations using conductive transfer only while the solid curve corresponds to calculations run with both conductive and radiative transfers. For panels (a) and (b), these curves are indistinguishable for the scale they are presented. The difference between these two curves is shown in gray (referred to right ordinates). Insets show the external power time profile applied to the system.

Finite element simulation of the coupler surface temperature history. (a)–(c) are corresponding to 2 *μ*s, 10 ns, CW laser heating, respectively. The dashed curve corresponds to calculations using conductive transfer only while the solid curve corresponds to calculations run with both conductive and radiative transfers. For panels (a) and (b), these curves are indistinguishable for the scale they are presented. The difference between these two curves is shown in gray (referred to right ordinates). Insets show the external power time profile applied to the system.

Finite element simulation of the Fe sample surface temperature history for the pulsed heating experiment of Ref. ^{ 17 } for different values of the peak laser power. The case of the symmetric double-sided heating is examined. Panels (a) and (b) correspond to the cases of constant emissivity and step-like emissivity decrease through the melting transition, respectively.

Finite element simulation of the Fe sample surface temperature history for the pulsed heating experiment of Ref. ^{ 17 } for different values of the peak laser power. The case of the symmetric double-sided heating is examined. Panels (a) and (b) correspond to the cases of constant emissivity and step-like emissivity decrease through the melting transition, respectively.

Comparison of the experimentally determined peak surface temperature of the Fe sample ^{ 17 } with the calculated results from finite element simulations for different peak laser powers.

Comparison of the experimentally determined peak surface temperature of the Fe sample ^{ 17 } with the calculated results from finite element simulations for different peak laser powers.

Finite element calculations for the surface temperature history of a Pt laser light absorber in H_{2} medium for different peak laser powers. The case of the symmetric double-sided heating is examined. The geometrical parameters and laser pulse parameters are from the experimental work of Ref. ^{ 34 } . The constant thermal conductivity and that increased by a factor of 10 at the melt line refer to the temperature dependence of this quantity for hydrogen, which was assumed in the calculations.

Finite element calculations for the surface temperature history of a Pt laser light absorber in H_{2} medium for different peak laser powers. The case of the symmetric double-sided heating is examined. The geometrical parameters and laser pulse parameters are from the experimental work of Ref. ^{ 34 } . The constant thermal conductivity and that increased by a factor of 10 at the melt line refer to the temperature dependence of this quantity for hydrogen, which was assumed in the calculations.

Finite element calculations of the surface temperature of the Pt laser light absorber as a function of the peak laser power. The constant thermal conductivity and that increased by a factor of 10 at the melt line refer to the temperature dependence of this quantity for hydrogen, which was assumed in the calculations.

Finite element calculations of the surface temperature of the Pt laser light absorber as a function of the peak laser power. The constant thermal conductivity and that increased by a factor of 10 at the melt line refer to the temperature dependence of this quantity for hydrogen, which was assumed in the calculations.

Finite element calculations of the surface temperature histories (T1 and T2 correspond to different sides of the sample) of a plate-like sample heated from one side (side 1) with a laser having a wave function time profile. The amplitude of the laser power modulation was 10% of the total applied laser power. The laser power was ramped up as shown in the inset to Fig. 3(c) . The results shown correspond to periods of time when the steady state is reached. The straight lines with the arrow illustrate the phase shift between the surface temperatures. Left and right panels correspond to two different values of the thermal conductivity of the sample.

Finite element calculations of the surface temperature histories (T1 and T2 correspond to different sides of the sample) of a plate-like sample heated from one side (side 1) with a laser having a wave function time profile. The amplitude of the laser power modulation was 10% of the total applied laser power. The laser power was ramped up as shown in the inset to Fig. 3(c) . The results shown correspond to periods of time when the steady state is reached. The straight lines with the arrow illustrate the phase shift between the surface temperatures. Left and right panels correspond to two different values of the thermal conductivity of the sample.

Finite element calculations of the phase shift (left panel) and amplitude ratios (right panel) of the surface temperatures of a plate-like sample heated from one side with a laser having a wave function time profile. The results are presented as a function of the dimensionless parameter *u*. The solid lines are guides to the eye for the calculations performed for the medium (Ar) with thermal conductivity K_{medium}(300 K) = 10 W/(m × K). The gray thick lines correspond to the results (Kelvin functions) for the classic Ångström method (for a cylindrical sample).

Finite element calculations of the phase shift (left panel) and amplitude ratios (right panel) of the surface temperatures of a plate-like sample heated from one side with a laser having a wave function time profile. The results are presented as a function of the dimensionless parameter *u*. The solid lines are guides to the eye for the calculations performed for the medium (Ar) with thermal conductivity K_{medium}(300 K) = 10 W/(m × K). The gray thick lines correspond to the results (Kelvin functions) for the classic Ångström method (for a cylindrical sample).

## Tables

Thermochemical parameters of materials used in model FE calculations (Fig. 2 ).

Thermochemical parameters of materials used in model FE calculations (Fig. 2 ).

Laser intensity parameters needed for heating of a sample surface to a peak temperature of 4000 K (one side) in FE calculations (Fig. 2 ).

Laser intensity parameters needed for heating of a sample surface to a peak temperature of 4000 K (one side) in FE calculations (Fig. 2 ).

Thermochemical parameters of materials used in FE calculations of the melting behavior of Fe in N_{2} medium at 18.8 GPa (Fig. 4 ).

Thermochemical parameters of materials used in FE calculations of the melting behavior of Fe in N_{2} medium at 18.8 GPa (Fig. 4 ).

Thermochemical parameters of materials used in FE calculations of the melting behavior of hydrogen (using a Pt laser light absorber) at 75 GPa (Fig. 6 ).

Thermochemical parameters of materials used in FE calculations of the melting behavior of hydrogen (using a Pt laser light absorber) at 75 GPa (Fig. 6 ).

Thermochemical parameters of materials used in FE calculations of the time-dependent temperature map in the DAC cavity, which occurs in response to a single-sided laser heating with a wave function time profile.

Thermochemical parameters of materials used in FE calculations of the time-dependent temperature map in the DAC cavity, which occurs in response to a single-sided laser heating with a wave function time profile.

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