Thermal Conductivity Measurements in Convergent Geometry

Yuan Ping | 17-FS-031


We assessed the feasibility of using a novel approach to measure thermal conductivity under previously inaccessible plasma conditions. The proposed thermal-conductivity experiments required a comprehensive analysis of multiple observables based on a relatively simple implosion stagnation model. The proposed model is designed to provide access to the time-dependent hot-spot-state variables in isobaric capsule implosions and a full suite of x-ray and neutron diagnostic quantities. The model is rapid enough to enable a numerical exploration of a large space of physics and state variables. Using this model, we can apply well-known machine learning and statistical methods to infer microphysics quantities that can be compared with state-of-the-art theory.

Background and Research Objectives

Thermal conductivity is a fundamental property of matter that is a necessary input in every hydrodynamic simulation code. It plays a key role in determining the instability growth rate at material interfaces (Hammel et al. 2010) and energy balance in inertial confinement fusion (ICF) implosions (Hurricane et al. 2016), as well as the thermal convection rate inside the Sun's convection zone (Brummel et al. 1995). However, experimental data are very scarce in the high-energy-density (HED) regime; therefore, all theoretical models remain untested and unvalidated for HED matter. We successfully measured thermal conductivity of Al (Ping et al. 2015, McKelvey et al. 2017) for the first time in a warm dense matter regime in planar geometry. A convergent geometry is required to reach higher temperatures, densities, and pressures.

Our goal was to develop a novel method to measure thermal conductivity in a hot spot created by a spherical ICF implosion. The radial profile of the hot spot is sensitive to thermal conductivity as a result of thermal conduction loss along the strong gradient. This measurement is enabled by a penumbral imaging technique (Bachmann et al. 2014) that provides the required spatial resolution. Preliminary simulations demonstrated the feasibility of such measurements. The absolute electron temperature profile can also be obtained by measurements in a range of x-ray energies (Kraus et al. 2014).

It is difficult to measure thermal conductivity experimentally because thermal conduction occurs only in the presence of a thermal gradient that is usually difficult to control. However, imploding shells set up a thermally dominated temperature profile, with a spatial shape that is strongly dependent on the thermal conductivity. Experiments that can constrain the spatial temperature profile can constrain the conductivity. In hydrodynamic simulations, we have found that the radial temperature profile of the hot spot strongly depends on thermal conductivity (i.e., different thermal conductivity leads to a different slope of the gradient). This is similar to the results obtained during earlier research (Le Pape et al. 2014) where the slope of a 200-μm hot spot is consistent with Spitzer conductivity for deuterium at 1 g/cc and 2 keV. We explored significantly higher densities where degeneracy and coupling become non-negligible. This requires much higher compression, hence much better spatial resolution. The x-ray penumbral imaging technique (Bachmann et al. 2016) enabled such measurements.

Using the heat conduction equation in spherical geometry, we were able to arrive at an analytical solution of the temperature radial profile (i.e., the radius of the hot spot). We were then able to confirm the analytical spatial profile and sensitivity to thermal conductivity using hydrodynamic simulations (see figure).




Left: Normalized synthetic image of the hot spot at 10 keV and three different thermal conductivities showing the sensitivity of the radial profile ( solid black plot line ) to thermal conductivity ( solid and dashed blue plot lines ). Right: Fitting of the simulated temperature profile from hydrodynamic simulations by the analytical solution at three snap shots. The good fits confirm the validity of the analytical model.




Impact on Mission

This project supports the NNSA goals to strengthen the science, technology, and engineering capabilities base and to expand and apply those capabilities to deal with broader national security challenges. Our research also supports the Laboratory's core competency in HED science by providing an innovative fundamental HED scientific capability at the National Ignition Facility (NIF) and OMEGA facility. Specifically, it supports the investigation of the effects of plasma transport properties. This project also supports the Laboratory's stockpile stewardship mission area to create more opportunities for innovation, enhance the vitality of HED experimental science, and maximize the usefulness of NIF for exploring HED physics under more extreme conditions.


Multiple penumbral images at different x-ray energy bands were required to measure the temperature radial profile. We tested the signal level in ride-along shots at the OMEGA laser facility and found that the hot spot was not bright enough for such measurements. Our NIF Discovery Science shot scheduled for July 2018 was deferred due to target issues. On October 29, 2018, this shot was finally conducted and produced excellent data in all five x-ray energy channels. More detailed data analysis currently is underway. Three technical papers are in preparation.

The rapid model of the stagnation states of the ICF implosions developed during this project has found applications in a current Laboratory project on machine learning. A follow-up Discovery Science proposal for NIF shots has been submitted; it will employ spatially and temporally resolved temperature measurements in the hot spot to test thermal conductivity models of hydrogen in unprecedented conditions.


Bachmann, B., et al. 2014. "Using Penumbral Imaging to Measure Micrometer Size Plasma Hot Spots in Gbar Equation of State Experiments on the National Ignition Facility." Review of Scientific Instruments 85(11): 11D614. doi: 10.1063/1.4891303.

——— . 2016. "Resolving Hot Spot Microstructure Using X-Ray Penumbral Imaging." Review of Scientific Instruments 87(11): 11E201. doi: 10.1063/1.4959161.

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Hammel, B. A., et al. 2010. "High-Mode Rayleigh-Taylor Growth in NIF Ignition Capsules." High Energy Density Physics 6(2): 171–78. doi: 10.1016/j.hedp.2009.12.005.

Hurricane, O., et al. 2016. "Inertially Confined Fusion Plasmas Dominated by Alpha-Particle Self-Heating." Nature Physics 12: 800–6. doi: 10.1038/nphys3720.

Kraus, D., et al. 2014. "X-Ray Continuum Emission Spectroscopy from Hot Dense Matter at Gbar Pressures." Review of Scientific Instruments 85(11): 11D606. doi: 10.1063/1.4890263.

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McKelvey, A., et al. 2017. "Thermal Conductivity Measurements of Proton-Heated Warm Dense Aluminum." Scientific Reports 7: 7015. doi: 10.1038/s41598-017-07173-0.

Ping, Y., et al. 2015. "Differential Heating: A Versatile Method for Thermal Conductivity Measurements in High-Energy-Density Matter." Physics of Plasmas 22(9): 092701. doi: 10.1063/1.4929797.