Mie–Gruneisen equation of state

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This article is based on the paper A.M. Krivtsov, V.A. Kuzkin, Derivation of Equations of State for Ideal Crystals of Simple Structure // Mech. Solids. 46 (3), 387-399 (2011)

Mie-Gruneisen equation of state

In high pressure physics it is usual to represent the total pressure [math]p[/math] in condensed matter as a sum of "cold" and "thermal" components:

[math]p = p_0 + p_T, ~~~~ p_T = p - p_0[/math]

The cold pressure, refereed to as the "cold curve" is caused by deformation of crystal lattice only. The thermal pressure is due to thermal motion of the atoms. In other words, the cold pressure is a function of volume only, while the thermal pressure also depends on thermal energy [math] E_T [/math]:

[math]p = p_0(V) + p_T(V,E_T)[/math]

The thermal energy is a part of the internal energy caused by the thermal motion of atoms. In the simplest case the thermal energy os equal to [math] c_V T [/math], where [math] c_V [/math] is the specific heat. In practice it is usually assumed that the dependence of pressure on thermal energy is linear:

[math] p = p_0(V) + \frac{\varGamma(V)}{V} E_T[/math]

The given equation is refereed to as Mie-Gruneisen equation of state (EOS). The function [math]\varGamma(V)[/math] is called Gruneisen function. The value [math] \varGamma_0 [/math] of Gruneisen function in undeformed configuration is called Gruneisen coefficient (or Gruneisen constant).

[math] \varGamma_0 = \varGamma(V_0)[/math]

Equation of state for perfect crystals with simple lattice

[math] p_0 = \frac{1}{2V_0d\theta^d}\sum_{k=1}^n N_k\varPhi_k A_k^2,~~~~\varGamma = -\frac{\sum_{k=1}^n N_k((d+2)\varPhi'_k A_k^2 + 2\varPhi''_k A_k^4 )}{d\sum_{k=1}^n N_k (d\varPhi_k +2\varPhi'_k A_k^2)} [/math]

where [math]k[/math] is the number of coordination sphere, [math]n[/math] is the number of coordination spheres, [math]N_k[/math] is the number of atoms bolonging to the [math]k[/math]-th coordination sphere, [math] A_k = \rho_k R \theta[/math] is the radius of coordination sphere , [math] \rho_k=A_k/A_1 [/math], [math]R[/math] is the radius of the first coordination sphere in undeformed configuration, [math]\varPhi^{(n)}_k = \varPhi^{(n)}(A_k^2)[/math].

Cold curve for Lennard-Jones, Mie, and Morse potentials

In the case of nierest neighbors interactions the cold curve for Lennard-Jones, Mie, and Morse potentials has the following simple form.

• Cold curve for Lennard-Jones potential:

[math] \varPi(r) =D\left[\left(\frac{a}{r}\right)^{12}-2\left(\frac{a}{r}\right)^{6}\right], ~~~~ p_0 = \frac{6MD}{dV_0\theta^{d}}(\theta^{-12}-\theta^{-6}) [/math]

• Cold curve for Mie potential:

[math] \varPi(r) =\frac{D}{n-m} \left[m\left(\frac{a}{r}\right)^{n}-n\left(\frac{a}{r}\right)^{m} \right], ~~~~ p_0 =\frac{m n MD}{2d(n-m)V_0\theta^{d}}\left(\theta^{-n}-\theta^{-m}\right) [/math]

• Cold curve for Morse potential:

[math] \varPi(r) = D\left[e^{2\alpha(a-r)}-2e^{\alpha(a-r)}\right], ~~~~ p_0 = \frac{\alpha a MD}{d V_0\theta^{d-1}} \left[e^{2\alpha a(1-\theta)}-e^{\alpha a(1-\theta)}\right] [/math]

Here [math]D[/math] is the bond energy, [math]a[/math] is the bond length, [math]\alpha[/math] is the parameter characterising the width of the potential well; [math]m, n[/math] are parameters of Mie potential.

Gruneisen constant for Lennard-Jones, Mie, and Morse potentials

The expression for Gruneisen parameter for crystalls with simple lattice and nearest-neighbor interactions in [math]d[/math]-dimmensional space has the form:

[math] \varGamma_0 = -\frac{1}{2d}\frac{\varPi'''(a)a^2 + (d-1)\left[\varPi''(a)a - \varPi'(a)\right]}{\varPi''(a)a + (d-1)\varPi'(a)} [/math]

where [math]\Pi[/math] is the interatomic potential, [math]a[/math] is the equlibrium ditance, [math]d[/math] is the space dimmensinality. The relation between Gruneisen constant and parameters of Lennard-Jones, Mie, and Morse potentials is presented in the table below.

Lattice	Dimmensionality	Lennard-Jones potential	Mie potential	Morse potential
Chain	[math] d=1 [/math]	[math]10\frac{1}{2} [/math]	[math]\frac{m+n+3}{2}[/math]	[math]\frac{3\alpha a}{2}[/math]
Triangular lattice	[math]d=2 [/math]	[math]5[/math]	[math] \frac{m+n+2}{4}[/math]	[math] \frac{3\alpha a - 1}{4}[/math]
ГЦК, ОЦК	[math]d=3 [/math]	[math]\frac{19}{6} [/math]	[math]\frac{n+m+1}{6}[/math]	[math]\frac{3\alpha a-2}{6}[/math]
"Hyperlattice"	[math]d=\infty[/math]	[math]-\frac{1}{2}[/math]	[math]-\frac{1}{2}[/math]	[math]-\frac{1}{2}[/math]
General formula	[math]d[/math]	[math]\frac{11}{d}-\frac{1}{2}[/math]	[math]\frac{m+n+4}{2d}-\frac{1}{2}[/math]	[math]\frac{3\alpha a + 1}{2d}-\frac{1}{2}[/math]

The expression for Gruneisen constant of 1D chain with Mie potential exactly coincides with the results of McDonald and Roy.

Gruneisen function for Lennard-Jones, Mie, and Morse potentials

In the case of nearest-neighbor interactions the Gruneisen function has the following form.

• Gruneisen function for Lennard-Jones potential:

[math] \varGamma = \frac{1}{d}\frac{4(8-d)\theta^{6}-7(14-d)}{(8-d)\theta^{6}-(14-d)}. [/math]

• Gruneisen function for Mie potential:

[math] \varGamma = \frac{1}{2d}\frac{(n+2)(n-d+2)\theta^{m-n}-(m+2)(m-d+2)}{(n-d+2)\theta^{m-n}-(m-d+2)}. [/math]

• Gruneisen function for Morse potential:

[math] \varGamma = \frac{1}{2d}\frac{e^{\alpha a(1-\theta)}\left(4\alpha^2a^2\theta^2-2d_1\alpha a \theta-d_1\right)-\left(\alpha^2 a^2\theta^2-d_1\alpha a\theta-d_1 \right)}{e^{\alpha a(1-\theta)}(2\alpha a\theta-d_1) -(\alpha a\theta-d_1)},~~ [/math] [math]d_1 = d-1,~~[/math] [math]\theta=(V/V_0)^{1/d}[/math]

Papers

• A.M. Krivtsov, V.A. Kuzkin, Derivation of Equations of State for Ideal Crystals of Simple Structure // Mech. Solids. 46 (3), 387-399 (2011).
• 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

Ссылки

• Статья про уравнение Ми-Грюнайзена в Википедии

• Mie–Gruneisen equation of state (Wikipedia