Heat-release equation of state and thermal conductivity of warm dense carbon by proton differential heating

Warm dense carbon is generated at 0.3–2.0 g/cc and 1–7 eV by proton heating. The release equation of state (EOS) after heating and thermal conductivity of warm dense carbon are studied experimentally in this regime using a Au/C dual-layer target to initiate a temperature gradient and two picosecond time-resolved diagnostics to probe the surface expansion and heat flow. Comparison between the data and simulations using various EOSs and thermal conductivity models is quantified with a statistical ${\chi }^2$ analysis. Out of seven EOS tables and five thermal conductivity models, only L9061 with the Lee-More model provides a probability above 50% to match all data.

Figure 1

(a) Experimental setup for proton heating of multilayer targets. Both the incident angle of the FDI beam and the SOP collection lens position are ${16}^{\circ }$ from target normal. The temporal resolution is 6 ps in SOP and 1 ps in FDI. (b) Proton energy spectra measured by the Thomson parabola spectrometer. (c) Measured carbon reflectivity vs time.

Figure 2

HYDRA simulation results to show effects of the EOS and ${\kappa }_t$. (a) and (b) Density and electron temperature profiles of the carbon layer at five delays for two EOS tables, L9061 and Sesame 7833, with the same ${\kappa }_t$ model Lee-More. (c) and (d) Similar profiles for two ${\kappa }_t$ models, Lee-More and GMS6, with the same EOS L9061. The location of the critical density is marked by black squares.

Figure 3

Comparison of data and HYDRA simulation outputs for two carbon thicknesses. (a) and (c) The time history of the temperature obtained from SOP. (b) and (d) The velocity obtained from FDI measurements. Three EOS tables, L9061 (blue), S7833 (black), L65 (green), and Lee-More thermal conductivity are used in these simulations.

Figure 4

Comparison of data and HYDRA simulation outputs using L9061 with three conductivity models: GMS6, Lee-More and Sesame 27834. (a) and (c) The time history of the temperature obtained from SOP. (b) and (d) The velocity obtained from FDI measurements.

Figure 5

(a) Velocity vs temperature near 20 ps. Two data points (78-nm- and 150-nm-thick carbon) and seven EOS tables are plotted for comparison. The effect of thermal conduction is shown as lines for the 150-nm case for three EOS tables that are closest to the data, L9061, S7833, and D7830. The five points along the line represent ${\kappa }_t$ models S27834, Purgatorio, Lee-More, $2\times$ Lee-More, and GMS6, respectively. (b) Thermal conductivity vs temperature predicted by five models at 1.5 g/cc. (c) Calculated thermal conductivity at 0.7 g/cc.

Figure 6

(a) Probabilities of the model matching the data calculated using the ${\chi }^2$ analysis with 40 points. The $P_{\mathrm{thres}}$ is marked as the red dashed line. (b) Thermal conductivity of carbon by $1.05\times$ Lee-More in unit of 100 W/(m K) as a function of density and temperature. Thermal conductivities in the range of $1.05\pm 0.15\times$ Lee-More are in agreement with the data within the experimental errors.

Figure 7

(a) Probabilities of L9061 with the Purgatorio conductivity model vs the scaling factor of the Purgatorio ${\kappa }_t$. (b) Probabilities of L9061 with $1.05\times$ Lee-More ${\kappa }_t$ as a function of the proton source multiplier.