Notes: Equation of State

Posted by Jingyi on March 21, 2022

Equation of State

Equation of state (EoS) describes the mathematical formulism relationships betweem pressure $P$, volume $V$ (or equivalently, specific density $\rho$) and temperature $T$. For a solid material, here are a series of important thermodynamic definitions:

Incompressibility $K_S=-V \left.\frac{\partial P}{\partial V}\right|_S\\ K_T=-V \left.\frac{\partial P}{\partial V}\right|_T$

Thermal expansivity $\alpha=\frac{1}{V} \left.\frac{\partial V}{\partial T}\right|_P$

Gruneisen parameter $\gamma = \frac{V}{C_V}\left.\frac{\partial P}{\partial T}\right|_V$

Therefore it brings the relation that $\frac{K_S}{K_T}=1+\alpha\gamma T$

For solid materials, the simple isothermal EoS can be derived as

$K = -\frac{dP}{d\ln V}=\frac{dP}{d\ln\rho}$

with constant $K=K_0$, we get $V=V_0 \exp\left(-\frac{P}{K_0}\right)$

Birch-Murnaghan (BM3) EoS

The finite strain EoS derived by Birch (1947), also referred as Birch-Murnaghan EoS, assumes the compression expressed as the Taylor series of the finite Eulerian strain $f$ given by $\frac{V_0}{V}=(1+2f)^{\frac{3}{2}}$

and the Helmholtz free energy of the solid is

$F(T)=a(T)f^2 + b(T)f^3 +c(T)f^4 + ...$

The third order Birch-Murnaghan (BM3) EoS is simply be derived as the third order expansion of $F(T)=a(T)f^2 + b(T)f^3$, which is

$P_{BM3}=\frac{3K_{0}}{2}\left[\left(\frac{V_0}{V}\right)^{\frac{7}{3}}-\left(\frac{V_0}{V}\right)^{\frac{75}{3}}\right]\left[1+\frac{3}{4}(K_{0}^\prime-4)\left[\left(\frac{V_0}{V}\right)^{\frac{2}{3}}-1\right]\right]$

or equally with $x=\frac{V}{V_0}$, the BM3 pressure and energy are given by

$P_{BM3}=\frac{3}{8}K_{0} \frac{x^{2/3}-1}{x^{10/3}}\left[3K_{0}^\prime x-16 x - 3 x^{1/3}(K_{0}^\prime-4) \right]$ $E_{BM3}=E_0 + \frac{9}{16}V_0 K_{0}\frac{(x^{2/3}-1)^2}{x^{7/3}} \left[x^{1/3}(K_{0}^\prime-4)-x(K_{0}^\prime-6)\right]$

which can be written as the following equation to keep consistent with the energy equation in the next part:

$E_{BM3}=E_0 + \frac{9}{16}V_0 K_{0}(\eta^2-1)^2\left[6+K_0^\prime(\eta^2-1)-4\eta^2\right]$ where $\eta=\left(\frac{V}{V_0}\right)^{1/3}=x^{1/3}$.

Vinet EoS

The compression can be expressed alternatively by the atomic potential representation. The Vinet pressure and energy are given by

$P_{Vinet} = 3K_{0}\frac{1-\eta}{\eta^2}\exp \left[-\frac{3}{2}(K_{0}^\prime-1)(\eta-1)\right]$ \begin{align} E_{Vinet} =& E_0 + \frac{4K_{0} V_0}{(K_{0}^\prime-1)^2} \\ &-\frac{2K_{0}V_0}{(K_{0}^\prime-1)^2}\left[3(K_{0}^\prime-1)(\eta-1)+2\right]\times\exp\left[-\frac{3}{2}(K_{0}^\prime-1)(\eta-1)\right] \end{align}

where $\eta=\left(\frac{V}{V_0}\right)^{1/3}=x^{1/3}$.

Memo

Before performing DFT calculations, we might not be able to know the energy vs volume relation but sometimes the pressure vs volume relation scatters or sets of parameters from previous experimental and theoratical studies. The pressure formula can help give proper volumes as DFT calculation input files.

After carrying out the DFT calculations, we usually fit the DFT energy with the computed volumes, giving the EoS parameters ($E_0$,$V_0$, $K_0$, $K_0^\prime$), and meanwhile we can obtain the pressure either by the derivative of free energy $P=\left.\frac{F}{V}\right|_T$ , or by inserting the fitting parameters into the pressure formula (if the functions of these equations have already been written and tested).