Mie–Gruneisen equation of state

Source

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.

[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

• 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]

Здесь [math]D[/math] - энергия связи, [math]a[/math] - длина связи, [math]\alpha[/math] - параметр, характеризующий ширину потенциальной ямы; [math]m, n[/math] - параметры потенциала Ми.

Коэффициент Грюнайзена для потенциалов Леннарда-Джонса, Ми, Морзе

Выражение для параметра Грюнайзена для идеальных кристаллов с парными взаимодействиями в пространстве размерности [math]d[/math] имеет вид:

[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]

где [math]\Pi[/math] - потенциал межатомного взаимодействия, [math]a[/math] - равновесное расстояние, [math]d[/math] - размерность пространства. Связь параметра Грюнайзена с параметрами потенциалов Леннарда-Джонса, Ми и Морзе представлена в таблице.

решетка	размерность пространства	Потенциал Леннарда-Джонса	Потенциал Ми	Потенциал Морзе
Цепочка	[math] d=1 [/math]	[math]10\frac{1}{2} [/math]	[math]\frac{m+n+3}{2}[/math]	[math]\frac{3\alpha a}{2}[/math]
Треугольная решетка	[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]
"Гиперрешетка"	[math]d=\infty[/math]	[math]-\frac{1}{2}[/math]	[math]-\frac{1}{2}[/math]	[math]-\frac{1}{2}[/math]
Общая формула	[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]

Функция Грюнайзена для потенциалов Леннарда-Джонса, Ми, Морзе

• Потенциал Леннарда-Джонса:

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

• Потенциал Ми

[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]

• Потенциал Морзе

[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]

Статьи

• Кривцов А. М., Кузькин В. А. Получение уравнения состояния идеальных кристаллов простой структуры // Механика твёрдого тела. — 2011. — № 3.

Ссылки

• Статья про уравнение Ми-Грюнайзена в Википедии

• Mie–Gruneisen equation of state