What Can We Learn from Hugoniot Temperature as a Function of Shock Velocity?

Shock-wave experiments traditionally rely on impact techniques, whereby measured shock velocity (US) can be related to material velocity (up), determined from the impact velocity (= 2up for a symmetric impact), and resulting in the empirically observed linear US-up equation of state: US = c0 + s up. Modern experiments relying on laser-driven compression have the advantage of reaching higher pressures than laboratory impact experiments, but up is typically not determined; instead, Hugoniot temperature (TH) and shock velocity are more readily measured. Assuming a linear US-up equation of state and that the Grüneisen parameter has the volume dependence g(V) = g0 (V/V0), measurements of the Hugoniot temperature as a function of shock velocity provide constraints on the specific heat along the Hugoniot CVH(US) = V0 f(US)[c0 g0 TH - s US dTH/dUS]-1 where the Walsh-Christian (1955) function f(US) = - (US - c0)2 US/(V0 s c0) = TH dSH/dVH gives the entropy change along the Hugoniot (subscripts 0 and H indicate zero-pressure and Hugoniot states, respectively). In this sense, TH(US) measurements are similar to calorimetry experiments. If specific heat and Grüneisen parameter are determined independently (e.g., from wave-velocity measurements and experiments on porous samples), the TH(US) analog to the linear US-up equation of state is TH(US) = {T0 exp(g0 /s) - ò[V0 c0 f(x)/(s x CV)] exp[c0 g0 /(s x)] dx} exp[- c0 g0 /(s US)] where the integration is from x = c0 to x = US. In addition, experiments can be considered with: 1) different initial volume, as in a porous sample; 2) different initial internal energy, as in a sample heated at constant volume; and 3) different initial volume and internal energy, as in a sample initially heated at ambient pressure. From these four initial states, we get four different Hugoniot curves, and can also consider the effect of phase transition latent heat. Temperature as a function of shock velocity may thus be benefit the analysis of melting and other phase transitions with small volume change and finite latent heat.