Supplementary Information: Linear dependence of surface expansion speed on initial plasma temperature in warm dense matter

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1 Supplementary Information: Linear dependence of surface expansion speed on initial plasma temperature in warm dense matter W. Bang, 1 B. J. Albright, 1 P. A. Bradley, 1 E. L. Vold, 1 J. C. Boettger, 1 and J. C. Fernández 1 1 Los Alamos National Laboratory, Los Alamos, NM, 87544, USA The linear dependence of surface expansion speed on initial plasma temperature appears to be a generic feature of warm dense matter rather than a unique characteristic of warm dense gold. Supplementary Figs. S1a,b show similar analyses for 10 m thick aluminum foils in the warm dense matter regime. The average surface expansion speed is plotted as a function of the initial plasma temperature from 1 to 100 ev in Supplementary Fig. S1a, and the expansion speed is shown as a function of the internal energy (corresponding to the same temperature range) in Supplementary Fig. S1b. The solid red curves in both figures indicate the power law fits to the data. In these simulations, we have examined the critical density surface for the 660 nm optical probe. As expected, Supplementary Fig. S1b shows a square-root dependence on the internal energy, while the surface expansion speed increases approximately linearly with temperature in Supplementary Fig. S1a. When a straight line is used to fit the data in Supplementary Fig. S1a, we get an R 2 value of According to the SESAME #3720 equation-of-state (EOS) table for aluminum, the temperature of aluminum can be approximated as T[eV]=0.50E[eV/atom] 0.70 in this regime, where E represents the internal energy per aluminum atom. Using this approximation in the power law fit in Supplementary Fig. S1b, we recover T 0.78 dependence and this is consistent with the power law fit in Supplementary Fig. S1a showing T dependence. 1

2 Supplementary Figure S1. Surface expansion speed of aluminum from RAGE simulations. (a) Average surface expansion speeds from RAGE 2D simulations are shown over plasma temperatures from 1 to 100 ev. The location of the critical density surface for 660 nm (hollow red circles) light is examined as a function of the initial temperature of warm dense aluminum in these simulations. (b) The expansion speed is shown here as a function of the internal energy of aluminum atom (corresponding to the same temperature range of 1 to 100 ev). 2

3 The surface expansion speed of an ideal plasma expanding into vacuum is known to be proportional to the sound speed, and we have calculated the sound speeds using the SESAME EOS tables in the warm dense matter regime. Supplementary Fig. S2 shows the expected sound speed in solid density gold as a function of temperature from 1 to 100 ev. These sound speeds are calculated along the isochore, which are different from sound speeds along an isobar, using the SESAME EOS #2700 table for gold. Interestingly, the sound speeds are also found to increase almost linearly with the plasma temperature in the warm dense matter regime (1 100 ev). This is consistent with the observed nearly linear dependence of the surface expansion speed on the initial temperature of warm dense gold foil. Again, the uncertainties in the SESAME EOS #2700 table can be larger than 10% in this regime, but we have not attempted to estimate these errors. In Supplementary Fig. S2, the deviations from linearity are well within the expected scatter in the calculated sound speeds due to trying to extract derivative quantities from a tabular EOS via interpolation using a rational function. Supplementary Figure S2. Sound speed in gold. The sound speed in solid density gold is calculated from 1 to 100 ev using the SESAME EOS #2700 table. The solid red line shows a linear fit to the data. 3

4 RAGE simulations in Fig. 3 predict that the expansion speed is insensitive to the initial foil thicknesses, and we have explained the physical origin of this in the discussion section. Therefore, we expect the same linear dependence of the surface expansion speed on the plasma temperature for 50 m thick and 100 m thick gold foils as in Fig. 4. In fact, Supplementary Figs. S3a,b confirm similar linear dependence, and nearly identical power law fits are retrieved for both cases. Specifically, the power law fits (solid red lines) to the 660 nm data indicate that the expansion speed can be approximated as 1.76 T[eV] 0.85 [ m/ns] for a 50 m thick gold foil and as 1.75 T[eV] 0.86 [ m/ns] for a 100 m thick gold foil, whereas v [ m/ns] = 1.75 T[eV] 0.86 for a 10 m thick gold foil in Fig. 4. The power law fits (solid green lines) to the 500 nm data indicate that the expansion speed can be estimated as 1.47 T[eV] 0.87 [ m/ns] for a 50 m thick gold foil and as 1.46 T[eV] 0.88 [ m/ns] for a 100 m thick gold foil, while it is estimated as 1.48 T[eV] 0.87 [ m/ns] for a 10 m thick gold foil in Fig. 4. The fits (solid violet lines) for the 400 nm data indicate that the expansion speed is 1.30 T[eV] 0.88 [ m/ns] for 10 m, 50 m, and 100 m thick gold foils. 4

5 Supplementary Figure S3. Nearly linear dependence of the surface expansion speed on the temperature of gold foils with different thicknesses. Surface expansion speeds from RAGE simulations are shown for three wavelengths over plasma temperatures from 1 to 100 ev for (a) 50 m thick and (b) 100 m thick gold foils. The locations of the critical density surfaces for 660 nm, 500 nm, and 400 nm light are examined as functions of the initial temperature of warm dense gold in these simulations. 5

6 In Supplementary Fig. S4, we compare ion number density profiles of an expanding gold foil at four different times: 0, t 0, 2t 0, and 3t 0. For simplicity, we assume linearly decaying ion number density profiles with the ion front propagating at a speed v fr toward vacuum and the rarefaction front propagating at the sound speed c s in the opposite direction. The electron number density is directly proportional to the ion number density satisfying the quasineutrality of the expanding plasma. (n e =<Z>n i with <Z> being the average charge state of the plasma) The arrows in the figure indicate how far particular density levels propagate toward vacuum during a fixed time interval of t 0, so their lengths are proportional to the expansion speeds of the corresponding density levels. For example, the red arrows drawn at the lowest density level represent the expansion speed of the critical density surface for a 660 nm light. It is clear from the figure that the red arrows are longer than the green arrows (corresponding to the expansion speed of the critical density surface for a 500 nm light) and violet arrows (corresponding to the expansion speed of the critical density surface for a 400 nm light). Note that the density profile remains unaffected for ions located deep (x<-3c s t 0 in the figure) inside the surface, and arrows of the same color have the same length representing a constant expansion speed of each density level. Supplementary Figure S4. Ion number density profiles at different times. The ion number density profiles are plotted at four different times from 0 to 3t 0 : at time 0 (dashed blue line), at t 0 (solid blue line), 2t 0 (dashed red line), and 3t 0 (solid red line). 6