Thermodynamic assessment for shock wave synthesis of transition metal silicides

Abstract

The initial state of the porous elemental powders is taken at 298°K. The final state is the fully reacted powders at the adiabatic temperature, T_(ad). Energy is input by the shock wave E_(shock) and by the negative enthalpy of formation ΔH_f of the silicide, E_(in) = E_(shock) – ΔH_f. The adiabatic process takes place on a closed system, and therefore T_(ad) depends only on the end points and not on the path of the process. We may therefore assume full reaction takes place first, followed by the input of the shock energy. Three regimes for T_(ad) are considered, where the heat of fusion of the silicide is given by ΔH_m. Define a function F(T_(upper)) as the integral of c_pdT from 298°K to temperature T_(upper), where c_p is the specific heat of the silicide given by c_p = A+ 10^(-3)BT+ 10^5CT^(-2) with constants A, B, and C. The input energy is converted to heat, and T_(ad) is obtained from: Regime I: T_(ad) < T_m, v = 0, and E_(in) = F(T_(ad)) where T_m is the silicide melting temperature, and v is the fraction of silicide melted. Regime II: T_(ad) = T_m, 0 ≤ v ≤ 1, for F(T_m) ≤ E_(in) ≤ F(T_m) +ΔH_m, and v = [E_(in) - F(T_m)]/C_(pl) where c_(pl) is the specif1c heat of the liquid silicide. Regime III: T_(ad) > T_m, v = 1, for E_(in) > F(T_m) + ΔH_m, and T_(ad) = T_m + [E_(in) - F(T_m) – ΔH_m]/c_(pl). The pressure dependence of the silicide melting points has not been determined and the zero pressure melting points were used in the calculations. (The silicon melting point decreases by 54°K/GPa (2), while that of the metals increases (3).) An iterative method of solution was employed to obtain T_(ad) in regime I. Calculations were made for the disilicides and 5:3 silicides of Cr, Mo, Nb, Re, Ta, Ti, V, W, and Zr. The silicide property values employed by Bhaduri et al. (1) were used, with the exception of ΔH_f values for Ti silicides (-88.48 J/mol for TiSi_2 and -288.8 J/mol for Ti_5Si_3) which were obtained from Ref. 4. Table I lists T_(ad) values at zero shock energy calculated here and in Ref. 1 for the 1:2 and 5:3 silicides. Table II gives calculated values of the shock energy required to reach the silicide melting point, the shock energy to melt the silicide, and the silicide melting point. Calculated values of Tad vs. shock energy are presented in Figs. 1 and 2.