Mie–Grüneisen equation of state

The Mie–Grüneisen equation of state is an equation of state that relates the pressure and 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–Grüneisen equation of state are in use.

The Grüneisen model can be expressed in the form

${\displaystyle \Gamma =V\left({\frac {dp}{de}}\right)_{V}}$

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

${\displaystyle p-p_{0}={\frac {\Gamma }{V}}(e-e_{0})}$

where ${\displaystyle p_{0}}$ and ${\displaystyle e_{0}}$ 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.

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]

A temperature-corrected version that is used in computational mechanics has the form^[6][7]: 61

${\displaystyle p={\frac {\rho _{0}C_{0}^{2}\chi \left[1-{\frac {\Gamma _{0}}{2}}\,\chi \right]}{\left(1-s\chi \right)^{2}}}+\Gamma _{0}E;\quad \chi :=1-{\cfrac {\rho _{0}}{\rho }}}$

where ${\displaystyle C_{0}}$ is the bulk speed of sound, ${\displaystyle \rho _{0}}$ is the initial density, ${\displaystyle \rho }$ is the current density, ${\displaystyle \Gamma _{0}}$ is Grüneisen's gamma at the reference state, ${\displaystyle s=dU_{s}/dU_{p}}$ is a linear Hugoniot slope coefficient, ${\displaystyle U_{s}}$ is the shock wave velocity, ${\displaystyle U_{p}}$ is the particle velocity, and ${\displaystyle E}$ is the internal energy per unit reference volume. An alternative form is

${\displaystyle p={\frac {\rho _{0}C_{0}^{2}(\eta -1)\left[\eta -{\frac {\Gamma _{0}}{2}}(\eta -1)\right]}{\left[\eta -s(\eta -1)\right]^{2}}}+\Gamma _{0}E;\quad \eta :={\cfrac {\rho }{\rho _{0}}}\,.}$

A rough estimate of the internal energy can be computed using

${\displaystyle E={\frac {1}{V_{0}}}\int C_{v}dT\approx {\frac {C_{v}(T-T_{0})}{V_{0}}}=\rho _{0}c_{v}(T-T_{0})}$

where ${\displaystyle V_{0}}$ is the reference volume at temperature ${\displaystyle T=T_{0}}$, ${\displaystyle C_{v}}$ is the heat capacity and ${\displaystyle c_{v}}$ is the specific heat capacity at constant volume. In many simulations, it is assumed that ${\displaystyle C_{p}}$ and ${\displaystyle C_{v}}$ are equal.

material	${\displaystyle \rho _{0}}$ (kg/m3)	${\displaystyle c_{v}}$ (J/kg-K)	${\displaystyle C_{0}}$ (m/s)	${\displaystyle s}$	${\displaystyle \Gamma _{0}}$ (${\displaystyle T<T_{1}}$)	${\displaystyle \Gamma _{0}}$ (${\displaystyle T>=T_{1}}$)	${\displaystyle T_{1}}$ (K)
Copper	8960	390	3933 [8]	1.5 [8]	1.99 [9]	2.12 [9]	700

From Grüneisen's model we have

${\displaystyle p-p_{0}={\frac {\Gamma }{V}}(e-e_{0})}$

(1)

where ${\displaystyle p_{0}}$ and ${\displaystyle e_{0}}$ are the pressure and internal energy at a reference state. The Hugoniot equations for the conservation of mass, momentum, and energy are

${\displaystyle {\begin{aligned}\rho _{0}U_{s}&=\rho (U_{s}-U_{p})\,,\\[1ex]p_{H}-p_{H0}&=\rho _{0}U_{s}U_{p}\,,\\[1ex]p_{H}U_{p}&=\rho _{0}U_{s}\left({\frac {U_{p}^{2}}{2}}+E_{H}-E_{H0}\right)\end{aligned}}}$

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

${\displaystyle {\frac {U_{p}}{U_{s}}}=1-{\frac {\rho _{0}}{\rho }}=1-{\frac {V}{V_{0}}}=:\chi \,.}$

Where we defined ${\displaystyle V=1/\rho }$, the specific volume (volume per unit mass).

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

${\displaystyle U_{s}=C_{0}+s\chi U_{s}\quad {\text{or}}\quad U_{s}={\frac {C_{0}}{1-s\chi }}\,.}$

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

${\displaystyle p_{H}=\rho _{0}\chi U_{s}^{2}={\frac {\rho _{0}C_{0}^{2}\chi }{(1-s\chi )^{2}}}\,.}$

Similarly, from the energy equation we have

${\displaystyle p_{H}\chi U_{s}={\tfrac {1}{2}}\rho \chi ^{2}U_{s}^{3}+\rho _{0}U_{s}E_{H}={\tfrac {1}{2}}p_{H}\chi U_{s}+\rho _{0}U_{s}E_{H}\,.}$

Solving for e_H, we have

${\displaystyle E_{H}={\tfrac {1}{2}}{\frac {p_{H}\chi }{\rho _{0}}}={\tfrac {1}{2}}p_{H}(V_{0}-V)}$

With these expressions for p_H and E_H, the Grüneisen model on the Hugoniot becomes

${\displaystyle p_{H}-p_{0}={\frac {\Gamma }{V}}\left({\frac {p_{H}\chi V_{0}}{2}}-e_{0}\right)\quad {\text{or}}\quad {\frac {\rho _{0}C_{0}^{2}\chi }{(1-s\chi )^{2}}}\left(1-{\frac {\chi }{2}}\,{\frac {\Gamma }{V}}\,V_{0}\right)-p_{0}=-{\frac {\Gamma }{V}}e_{0}\,.}$

If we assume that Γ/V = Γ₀/V₀ and note that ${\displaystyle p_{0}=-de_{0}/dV}$, we get

${\displaystyle {\frac {\rho C_{0}^{2}\chi }{(1-s\chi )^{2}}}\left(1-{\frac {\Gamma _{0}\chi }{2}}\right)+{\frac {de_{0}}{dV}}+{\frac {\Gamma _{0}}{V_{0}}}e_{0}=0\,.}$

(2)

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

${\displaystyle {\begin{aligned}e_{0}={\frac {\rho C_{0}^{2}V_{0}}{2s^{4}}}{\Biggl [}&\exp(\Gamma _{0}\chi )\left({\tfrac {\Gamma _{0}}{s}}-3\right)s^{2}-{\frac {\left[{\tfrac {\Gamma _{0}}{s}}-(3-s\chi )\right]s^{2}}{1-s\chi }}+\\&\exp \left[-{\tfrac {\Gamma _{0}}{s}}(1-s\chi )\right]\left(\Gamma _{0}^{2}-4\Gamma _{0}s+2s^{2}\right)\left({\text{Ei}}\left[{\tfrac {\Gamma _{0}}{s}}(1-s\chi )\right]-{\text{Ei}}\left[{\tfrac {\Gamma _{0}}{s}}\right]\right){\Biggr ]}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}p_{0}=-{\frac {de_{0}}{dV}}={\frac {\rho C_{0}^{2}}{2s^{4}(1-\chi )}}{\Biggl [}&{\frac {s}{(1-s\chi )^{2}}}{\Bigl (}-\Gamma _{0}^{2}(1-\chi )(1-s\chi )+\Gamma _{0}[s\{4(\chi -1)\chi s-2\chi +3\}-1]\\&\qquad \qquad \quad -\exp(\Gamma _{0}\chi )[\Gamma _{0}(\chi -1)-1](1-s\chi )^{2}(\Gamma _{0}-3s)+s[3-\chi s\{(\chi -2)s+4\}]{\Bigr )}\\&-\exp \left[-{\tfrac {\Gamma _{0}}{s}}(1-s\chi )\right]\left[\Gamma _{0}(\chi -1)-1\right]\left(\Gamma _{0}^{2}-4\Gamma _{0}s+2s^{2}\right)\left({\text{Ei}}\left[{\tfrac {\Gamma _{0}}{s}}(1-s\chi )\right]-{\text{Ei}}\left[{\tfrac {\Gamma _{0}}{s}}\right]\right){\Biggr ]}\,.\end{aligned}}}$

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

${\displaystyle e_{0}(V)=A+B\chi (V)+C\chi ^{2}(V)+D\chi ^{3}(V)+\cdots }$

and

${\displaystyle p_{0}(V)=-{\frac {de_{0}}{dV}}=-{\frac {de_{0}}{d\chi }}\,{\frac {d\chi }{dV}}={\frac {1}{V_{0}}}\,(B+2C\chi +3D\chi ^{2}+\cdots )\,.}$

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

${\displaystyle p={\frac {1}{V_{0}}}\,(B+2C\chi +3D\chi ^{2}+\cdots )+{\frac {\Gamma _{0}}{V_{0}}}\left[e-(A+B\chi +C\chi ^{2}+D\chi ^{3}+\cdots )\right]\,.}$

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

${\displaystyle p={\frac {1}{V_{0}}}\left[2C\chi \left(1-{\tfrac {\Gamma _{0}}{2}}\chi \right)+3D\chi ^{2}\left(1-{\tfrac {\Gamma _{0}}{3}}\chi \right)+\cdots \right]+\Gamma _{0}E}$

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

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

${\displaystyle C={\frac {\rho _{0}C_{0}^{2}V_{0}}{2(1-s\chi )^{2}}}\,.}$

Then we get the following expression for p:

${\displaystyle p={\frac {\rho _{0}C_{0}^{2}\chi }{(1-s\chi )^{2}}}\left(1-{\tfrac {\Gamma _{0}}{2}}\chi \right)+\Gamma _{0}E\,.}$

This is the commonly used first-order Mie–Grüneisen equation of state.^{[citation needed]}

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