Four-Parameter Universal Equation of State
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摘要: 多种材料的拉格朗日体模量都符合对压强展开到二阶项的关系。以此为基础，建立了一种四参数物态方程，比三参数物态方程适用压力宽、拟合精度高、外推性能好。方程参数关联一、二、三阶导数，根据Hugoniot方程与等熵方程的导数关系，可以检验动高压与静高压实验数据是否相互适合。方程以统一形式表述冲击、等熵和等温压缩状态，适用于多种材料、全压力区，是一种普适物态方程。公式简单、运算方便，克服了传统的四参数物态方程结构复杂、公式冗长以及参数间高度关联而失去实用意义的缺陷。Abstract: Based on the fact that the Lagrangian bulk moduli of many materials can be expressed as a second order expansion with respect to pressure, a four-parameter equation of state has been proposed which, in comparison with the three-parameter equation of state, can be applied to a wider pressure range and has higher fitting accuracy and good extrapolation behavior.The parameters in the equation are dependent on the first, second and third order derivatives, hence, by using the derivative relations of the Hugoniot and isentropic equations, one can verify if the dynamic high-pressure experimental data match the static high-pressure experimental data.The equation describes shock, isentropic and isothermal compression states in a unified form, is suitable for a variety of materials in a wide pressure range, and therefore can be considered as a universal equation of state.Being simple and easy to operate, our equation overcomes the weakness of conventional four-parameter equations of state which are of no practical use due to their complicated structure, lengthy formula, and high correlation between parameters.
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Key words:
- Lagrangian bulk modulus /
- universal equation of state /
- unified form
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图 2 不同方程冲击压缩线外推比较
Figure 2. Comparison of the extrapolation Hugniot with different equations
图 3 不同方程等温压缩线外推比较
Figure 3. Comparison of the extrapolation isothermal compession cuver with different equations
图 4 (6) 式向负压区外推线
Figure 4. Extrapolation of Eq.(6) to the region of the negative pressure
表 1 10种金属元素物态方程参数及其拉格朗日体模量参数
Table 1. Parameters of the equation of states and Lagrange bulk modulus of the ten metals
Metal Parameters of the equation of states Fitting parameters of the Lagrange bulk modulus ρ0/(g/cm3) c0/(km/s) λ B0/(GPa) b a/(GPa-1) Li 0.530 4.580 1.150 10.525 5.017 0.067 6 Mo 10.208 5.140 1.220 247.921 5.450 0.002 9 W 19.200 4.040 1.230 286.548 5.515 0.002 5 Ag 10.490 3.270 1.550 36.865 8.669 0.006 5 Ta 16.656 3.430 1.190 173.261 5.259 0.004 1 Metal ρ0K/(g/cm3) K0 K0′ B0/(GPa) b a/(GPa-1) Na 1.017 7.352 3.946 7.312 4.921 0.051 0 Mo 10.235 271.255 3.878 269.697 4.860 0.001 4 Pt 21.574 286.467 4.879 285.469 5.752 0.001 0 Au 19.433 185.132 5.146 184.708 5.981 0.001 4 Pd 12.080 196.744 5.201 196.263 6.031 0.001 3 
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表 2 (6) 式与不同方程对数据的拟合比较
Table 2. Comparison of the fitting using Eq.(6) and others
Metal Eq.(6) Eq.(7) p/(GPa) q s C2 R2 c0/(km/s) λ λ′ C2 R2 Cu-pH -24.860 0.046 5.530 11.310 0.999 2 4.113 1.399 0.0159 12.038 0.999 2 Fe-pH -10.427 0.002 8.675 22.525 0.998 1 3.660 1.806 -0.041 21.515 0.998 2 Eq.(4) B0/(GPa) B0′ C2 R2 Cu-pT -17.858 -0.024 7.237 0.877 0.999 3 131.701 5.332 0.913 0.999 2 Fe-pT -21.845 0.008 6.617 17.974 0.998 5 139.192 6.365 17.861 0.998 5
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