Mie–Gruneisen equation of state

The Mie-Grüneisen equation of state is a relation between the pressure and the volume of a solid at a given temperature.^[1]^[2] It is used to determine the pressure in a shock-compressed solid. The Mie-Grüneisen relation is a special form of the Grüneisen model which describes the effect that changing the volume of a crystal lattice has on its vibrational properties. Several variations of the Mie–Gruneisen equation of state are in use.

The Grüneisen model can be expressed in the form

\Gamma = V \left(\frac{dp}{de}\right)_V

where V is the volume, p is the pressure, e is the internal energy, and Γ is the Grüneisen parameter which represents the thermal pressure from a set of vibrating atoms. If we assume that Γ is independent of p and e, we can integrate Grüneisen's model to get

p - p_0 = \frac{\Gamma}{V} (e - e_0)

where p₀ and e₀ are the pressure and internal energy at a reference state usually assumed to be the state at which the temperature is 0K. In that case p₀ and e₀ are independent of temperature and the values of these quantities can be estimated from the Hugoniot equations. The Mie-Grüneisen equation of state is a special form of the above equation.

History

Gustav Mie, in 1903, developed an intermolecular potential for deriving high-temperature equations of state of solids.^[3] In 1912 Eduard Grüneisen extended Mie's model to temperatures below the Debye temperature at which quantum effects become important.^[4] Grüneisen's form of the equations is more convenient and has become the usual starting point for deriving Mie-Grüneisen equations of state.^[5]

Expressions for the Mie-Grüneisen equation of state

A temperature-corrected version that is used in computational mechanics has the form^[6] (see also,^[7] p. 61)

p = \frac{\rho_0 C_0^2 \chi \left[1 - \frac{\Gamma_0}{2}\,\chi\right]} {\left(1 - s\chi\right)^2} + \Gamma_0 E;\quad \chi := 1-\cfrac{\rho_0}{\rho}

where $C_0$ is the bulk speed of sound, $\rho_0$ is the initial density, $\rho$ is the current density, $\Gamma_0$ is Gruneisen's gamma at the reference state, $s = dU_s/dU_p$ is a linear Hugoniot slope coefficient, $U_s$ is the shock wave velocity, $U_p$ is the particle velocity, and $E$ is the internal energy per unit reference volume. An alternative form is

p = \frac{\rho_0 C_0^2 (\eta -1) \left[\eta - \frac{\Gamma_0}{2}(\eta-1)\right]} {\left[\eta - s(\eta-1)\right]^2} + \Gamma_0 E;\quad \eta := \cfrac{\rho}{\rho_0} \,.

A rough estimate of the internal energy can be computed using

E = \frac{1}{V_0} \int C_v dT \approx \frac{C_v (T-T_0)}{V_0} = \rho_0 c_v (T-T_0)

where $V_0$ is the reference volume at temperature $T = T_0$ , $C_v$ is the heat capacity and $c_v$ is the specific heat capacity at constant volume. In many simulations, it is assumed that $C_p$ and $C_v$ are equal.

Parameters for various materials

material	$\rho_0$ (kg/m³)	$c_v$ (J/kg-K)	$C_0$ (m/s)	$s$	$\Gamma_0$ ( $T < T_1$ )	$\Gamma_0$ ( $T >= T_1$ )	T_1 (K)
Copper	8960	390	3933 ^[8]	1.5 ^[8]	1.99 ^[9]	2.12 ^[9]	700

Gruneisen constant for perfect crystals with pair interactions

The expression for Gruneisen constant of a perfect crystal with pair interactions in $d$ -dimmensional space has the form:^[10]

$\Gamma_0 = -\frac{1}{2d}\frac{\Pi'''(a)a^2 + (d-1)\left[\Pi''(a)a - \Pi'(a)\right]}{\Pi''(a)a + (d-1)\Pi'(a)},$

where $\Pi$ is the interatomic potential, $a$ is the equilibrium distance, $d$ is the space dimensionality. Relations between the Gruneisen constant and parameters of Lennard-Jones, Morse, and Mie potentials are presented in the table below.

Lattice	Dimensionality	Lennard-Jones potential	Mie Potential	Morse potential
Chain	$d=1$	$10\frac{1}{2}$	$\frac{m+n+3}{2}$	$\frac{3\alpha a}{2}$
Triangual lattice	$d=2$	$5$	$\frac{m+n+2}{4}$	$\frac{3\alpha a - 1}{4}$
FCC, BCC	$d=3$	$\frac{19}{6}$	$\frac{n+m+1}{6}$	$\frac{3\alpha a-2}{6}$
"Hyperlattice"	$d=\infty$	$-\frac{1}{2}$	$-\frac{1}{2}$	$-\frac{1}{2}$
General formula	$d$	$\frac{11}{d}-\frac{1}{2}$	$\frac{m+n+4}{2d}-\frac{1}{2}$	$\frac{3\alpha a + 1}{2d}-\frac{1}{2}$

The expression for Gruneisen constant of 1D chain with Mie potential exactly coincides with the results of McDonald and Roy^[11] Using the relation between Gruneisen parameter and interatomic potential one can derive simple necessary and sufficient condition for Negative Thermal Expansion in perfect crystals with pair interactions:

$\Pi'''(a)a > -(d-1)\Pi''(a),$

See this page for more details.

Derivation of the equation of state

From Grüneisen's model we have

(1) \qquad p - p_0 = \frac{\Gamma}{V} (e - e_0)

where p₀ and e₀ are the pressure and internal energy at a reference state. The Hugoniot equations for the conservation of mass, momentum, and energy are

\rho_0 U_s = \rho (U_s - U_p) ~~, \quad p_H - p_{H0} = \rho_0 U_s U_p \quad \text{and} \quad p_H U_p = \rho_0 U_s \left(\frac{U_p^2}{2} + E_H - E_{H0}\right)

where ρ₀ is the reference density, ρ is the density due to shock compression, p_H is the pressure on the Hugoniot, E_H is the internal energy per unit mass on the Hugoniot, U_s is the shock velocity, and U_p is the particle velocity. From the conservation of mass, we have

\frac{U_p}{U_s} = 1 - \frac{\rho_0}{\rho} = 1 - \frac{V}{V_0} =: \chi \,.

For many materials U_s and U_p are linearly related, i.e., U_s = C₀ + s U_p where C₀ and s depend on the material. In that case, we have

U_s = C_0 + s\chi U_s \quad \text{or} \quad U_s = \frac{C_0}{1 - s\chi} \,.

The momentum equation can then be written (for the principal Hugoniot where p_H0 is zero) as

p_H = \rho_0 \chi U_s^2 = \frac{\rho C_0^2 \chi}{(1 - s\chi)^2} \,.

Similarly, from the energy equation we have

p_H \chi U_s = \tfrac{1}{2} \rho \chi^2 U_s^3 + \rho_0 U_s E_H = \tfrac{1}{2} p_H \chi U_s + \rho_0 U_s E_H \,.

Solving for e_H, we have

E_H = \tfrac{1}{2} \frac{p_H \chi}{\rho_0} = \tfrac{1}{2} \frac{p_H \chi V_0}{\rho V} \quad \text{or} \quad e_H = \tfrac{1}{2} p_H \chi V_0

where e_H is the total internal energy. With these expressions for p_H and e_H, the Grüneisen model on the Hugoniot becomes

p_H - p_0 = \frac{\Gamma}{V} \left(\frac{p_H \chi V_0}{2} - e_0\right) \quad \text{or} \quad \frac{\rho C_0^2 \chi}{(1 - s\chi)^2}\left(1 - \frac{\chi}{2}\,\frac{\Gamma}{V}\,V_0\right) - p_0 = -\frac{\Gamma}{V} e_0 \,.

If we assume that Γ/V = Γ₀/V₀ and note that $p_0 = -d e_0/d V$ , we get

(2) \qquad \frac{\rho C_0^2 \chi}{(1 - s\chi)^2}\left(1 - \frac{\Gamma_0\chi}{2}\right) + \frac{d e_0}{d V} + \frac{\Gamma_0}{V_0} e_0 = 0\,.

The above ordinary differential equation can be solved for e₀ with the initial condition e₀ = 0 when V = V₀ (χ = 0). The exact solution is

\begin{align} e_0 = \frac{\rho C_0^2 V_0}{2 s^4} \Biggl[&\exp(\Gamma_0\chi) (\tfrac{\Gamma_0}{s} - 3 ) s^2 - \frac{ [\tfrac{\Gamma_0}{s} - (3 - s\chi)]s^2}{1 - s\chi} + \\ & \exp\left[-\tfrac{\Gamma_0}{s} (1-s\chi)\right] (\Gamma_0^2 - 4 \Gamma_0 s + 2 s^2) (\text{Ei}[\tfrac{\Gamma_0}{s} (1-s\chi )] - \text{Ei}[\tfrac{\Gamma_0}{s}]) \Biggr] \end{align}

where Ei[z] is the exponential integral. The expression for p₀ is

\begin{align} p_0 = -\frac{de_0}{dV} = \frac{\rho C_0^2}{2s^4(1-\chi)} \Biggl[& \frac{s}{(1 - s\chi)^2} \Bigl (- \Gamma_0^2(1 - \chi)(1 -s\chi) + \Gamma_0 [s \{4 (\chi-1) \chi s-2 \chi+3\}-1] \\ & - \exp(\Gamma_0\chi)[\Gamma_0(\chi-1) -1](1-s\chi)^2(\Gamma_0-3s) + s [3-\chi s \{(\chi-2) s+4\}]\Bigr) \\ & - \exp\left[-\tfrac{\Gamma_0}{s} (1-s\chi)\right][\Gamma_0(\chi-1) - 1](\Gamma_0^2 - 4 \Gamma_0 s + 2 s^2)(\text{Ei}[\tfrac{\Gamma_0}{s} (1-s\chi )] - \text{Ei}[\tfrac{\Gamma_0}{s}]) \Biggr] \,. \end{align}

Plots of e₀ and p₀ for copper as a function of χ.

For commonly encountered compression problems, an approximation to the exact solution is a power series solution of the form

e_0(V) = A + B \chi(V) + C \chi^2(V) + D \chi^3(V) + \dots

and

p_0(V) = -\frac{de_0}{dV} = -\frac{de_0}{d\chi}\,\frac{d\chi}{dV} = \frac{1}{V_0}\,(B + 2C\chi + 3D\chi^2 + \dots) \,.

Substitution into the Grüneisen model gives us the Mie-Grüneisen equation of state

p = \frac{1}{V_0}\,(B + 2C\chi + 3D\chi^2 + \dots) + \frac{\Gamma_0}{V_0} \left[e - (A + B \chi + C \chi^2 + D \chi^3 + \dots ) \right] \,.

If we assume that the internal energy e₀ = 0 when V = V₀ (χ = 0) we have A = 0. Similarly, if we assume p₀ = 0 when V = V₀ we have B = 0. The Mie-Grüneisen equation of state can then be written as

p = \frac{1}{V_0}\left[2C\chi \left(1-\tfrac{\Gamma_0}{2}\chi\right) + 3D\chi^2\left(1 -\tfrac{\Gamma_0}{3}\chi\right) + \dots\right] + \Gamma_0 E

where E is the internal energy per unit reference volume. Several forms of this equation of state are possible.

Comparison of exact and first-order Mie-Grüneisen equation of state for copper.

If we take the first-order term and substitute it into equation (2), we can solve for C to get

C = \frac{\rho C_0^2 V_0}{2(1-s\chi)^2} \,.

Then we get the following expression for p :

p = \frac{\rho C_0^2 \chi}{(1-s\chi)^2} \left(1-\tfrac{\Gamma_0}{2}\chi\right) + \Gamma_0 E \,.

This is the commonly used first-order Mie-Grüneisen equation of state.

References

↑ Roberts, J. K., & Miller, A. R. (1954). Heat and thermodynamics (Vol. 4). Interscience Publishers.
↑ Burshtein, A. I. (2008). Introduction to thermodynamics and kinetic theory of matter. Wiley-VCH.
↑ Mie, G. (1903) "Zur kinetischen Theorie der einatomigen Körper." Annalen der Physik 316.8, p. 657-697.
↑ Grüneisen, E. (1912). Theorie des festen Zustandes einatomiger Elemente. Annalen der Physik, 344(12), 257-306.
↑ Lemons, D. S., & Lund, C. M. (1999). Thermodynamics of high temperature, Mie–Gruneisen solids. American Journal of Physics, 67, 1105.
↑ Zocher, M.A.; Maudlin, P.J. (2000), "An evaluation of several hardening models using Taylor cylinder impact data", Conference: COMPUTATIONAL METHODS IN APPLIED SCIENCES AND ENGINEERING, BARCELONA (ES), 09/11/2000--09/14/2000, retrieved 2009-05-12
↑ Wilkins, M.L. (1999), Computer simulation of dynamic phenomena, retrieved 2009-05-12
↑ 8.0 8.1 Mitchell, A.C.; Nellis, W.J. (1981), "Shock compression of aluminum, copper, and tantalum", Journal of Applied Physics 52 (5): 3363, Bibcode:1981JAP....52.3363M, doi:10.1063/1.329160, retrieved 2009-05-12
↑ 9.0 9.1 MacDonald, R.A.; MacDonald, W.M. (1981), "Thermodynamic properties of fcc metals at high temperatures", Physical Review B 24 (4): 1715–1724, Bibcode:1981PhRvB..24.1715M, doi:10.1103/PhysRevB.24.1715
↑ Krivtsov, A.M.; Kuzkin, V.A. (2011), "Derivation of Equations of State for Ideal Crystals of Simple Structure", Mechanics of Solids 46 (3): 387–399, doi:10.3103/S002565441103006X
↑ MacDonald, D. K. C.; Roy, S.K. (1955), "Vibrational Anharmonicity and Lattice Thermal Properties. II", Phys. Rev. 97: 673–676, doi:10.1103/PhysRev.97.673