Mie–Gruneisen equation of state — различия между версиями

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(Cold curve for Lennard-Jones, Mie, and Morse potentials)
 
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[[Уравнение состояния Ми-Грюнайзена | Русская версия]]
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== Source ==
 
== Source ==
This article is based on the paper '''A.M. Krivtsov, V.A. Kuzkin, [[Медиа: Krivtsov_2011_MechSol.pdf | Derivation of Equations of State for Ideal Crystals of Simple Structure]] // Mech. Solids. 46 (3), 387-399 (2011))'''
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This article is based on the paper  
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* Krivtsov A.M., Kuzkin V.A. '''Derivation of Equations of State for Ideal Crystals of Simple Structure''' // ''Mech. Solids.'' 46 (3), 387-399 (2011) (Download pdf: [[Медиа:Krivtsov_2011_MechSol.pdf‎|529 Kb]])
  
 
== Mie-Gruneisen equation of state ==
 
== Mie-Gruneisen equation of state ==
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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:
 
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:
  
Строка 15: Строка 21:
 
<math> p = p_0(V) + \frac{\varGamma(V)}{V} E_T</math>
 
<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'''.  
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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>
 
<math> \varGamma_0 = \varGamma(V_0)</math>
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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>.
 
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, Mie, and Morse potentials ==
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  \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})
 
  \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>
 
</math>
 
  
 
* '''Cold curve for Mie potential:'''
 
* '''Cold curve for Mie potential:'''
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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.
 
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.
  
== Коэффициент Грюнайзена для потенциалов Леннарда-Джонса, Ми, Морзе ==  
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== Gruneisen constant for Lennard-Jones, Mie, and Morse potentials ==  
  
Выражение для параметра Грюнайзена для идеальных кристаллов с парными взаимодействиями в пространстве размерности <math>d</math> имеет вид:
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The expression for Gruneisen parameter for crystalls with simple lattice and nearest-neighbor interactions in <math>d</math>-dimmensional space has the form:
  
 
<math>
 
<math>
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</math>
 
</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.
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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.
  
 
{|class="wikitable"
 
{|class="wikitable"
 
|-
 
|-
!решетка
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!Lattice
!размерность пространства
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!Dimmensionality
!Потенциал Леннарда-Джонса
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!Lennard-Jones potential
!Потенциал Ми
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!Mie potential
!Потенциал Морзе
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!Morse potential
 
|-
 
|-
| Цепочка
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| Chain
 
! <math> d=1 </math>
 
! <math> d=1 </math>
 
! <math>10\frac{1}{2} </math>
 
! <math>10\frac{1}{2} </math>
Строка 77: Строка 81:
 
! <math>\frac{3\alpha a}{2}</math>
 
! <math>\frac{3\alpha a}{2}</math>
 
|-
 
|-
| Треугольная решетка
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| Triangular lattice
 
!<math>d=2 </math>
 
!<math>d=2 </math>
 
! <math>5</math>
 
! <math>5</math>
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! <math>\frac{3\alpha a-2}{6}</math>
 
! <math>\frac{3\alpha a-2}{6}</math>
 
|-
 
|-
| "Гиперрешетка"
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| "Hyperlattice"
 
! <math>d=\infty</math>
 
! <math>d=\infty</math>
 
! <math>-\frac{1}{2}</math>
 
! <math>-\frac{1}{2}</math>
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! <math>-\frac{1}{2}</math>
 
! <math>-\frac{1}{2}</math>
 
|-
 
|-
| Общая формула
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| General formula
 
! <math>d</math>
 
! <math>d</math>
 
! <math>\frac{11}{d}-\frac{1}{2}</math>
 
! <math>\frac{11}{d}-\frac{1}{2}</math>
Строка 103: Строка 107:
 
|}
 
|}
  
== Функция Грюнайзена для потенциалов Леннарда-Джонса, Ми, Морзе ==  
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The expression for Gruneisen constant of 1D chain with Mie potential exactly coincides with the results of McDonald and Roy.
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== Gruneisen function for Lennard-Jones, Mie, and Morse potentials ==  
  
* Потенциал Леннарда-Джонса:
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In the case of nearest-neighbor interactions the Gruneisen function has the following form.
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* '''Gruneisen function for Lennard-Jones potential:'''
 
<math>
 
<math>
 
   \varGamma = \frac{1}{d}\frac{4(8-d)\theta^{6}-7(14-d)}{(8-d)\theta^{6}-(14-d)}.
 
   \varGamma = \frac{1}{d}\frac{4(8-d)\theta^{6}-7(14-d)}{(8-d)\theta^{6}-(14-d)}.
 
</math>
 
</math>
  
 
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* '''Gruneisen function for Mie potential:'''
* Потенциал Ми
 
 
<math>
 
<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)}.
 
     \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>
 
</math>
  
 
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* '''Gruneisen function for Morse potential:'''
* Потенциал Морзе
 
 
<math>
 
<math>
 
\varGamma = \frac{1}{2d}\frac{e^{\alpha a(1-\theta)}\left(4\alpha^2a^2\theta^2-2d_1\alpha a
 
\varGamma = \frac{1}{2d}\frac{e^{\alpha a(1-\theta)}\left(4\alpha^2a^2\theta^2-2d_1\alpha a
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<math>d_1 = d-1,~~</math> <math>\theta=(V/V_0)^{1/d}</math>
 
<math>d_1 = d-1,~~</math> <math>\theta=(V/V_0)^{1/d}</math>
  
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== Papers ==
 +
* Krivtsov A.M., Kuzkin V.A. '''Derivation of Equations of State for Ideal Crystals of Simple Structure''' // ''Mech. Solids.'' 46 (3), 387-399 (2011) (Download pdf: [[Медиа:Krivtsov_2011_MechSol.pdf‎|529 Kb]])
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* 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
  
 
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== Links ==
 
 
== Статьи ==
 
* Кривцов А. М., Кузькин В. А. Получение уравнения состояния идеальных кристаллов простой структуры // Механика твёрдого тела. — 2011. — № 3.
 
 
 
== Ссылки ==
 
  
 
* [http://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D1%8F_%D0%9C%D0%B8_%E2%80%94_%D0%93%D1%80%D1%8E%D0%BD%D0%B0%D0%B9%D0%B7%D0%B5%D0%BD%D0%B0 Статья про уравнение Ми-Грюнайзена в Википедии]
 
* [http://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D1%8F_%D0%9C%D0%B8_%E2%80%94_%D0%93%D1%80%D1%8E%D0%BD%D0%B0%D0%B9%D0%B7%D0%B5%D0%BD%D0%B0 Статья про уравнение Ми-Грюнайзена в Википедии]
  
* [http://en.wikipedia.org/wiki/Mie%E2%80%93Gruneisen_equation_of_state  Mie–Gruneisen equation of state]
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* [http://en.wikipedia.org/wiki/Mie%E2%80%93Gruneisen_equation_of_state  Mie–Gruneisen equation of state (Wikipedia]

Текущая версия на 00:54, 16 декабря 2013

Русская версия

Source[править]

This article is based on the paper

  • Krivtsov A.M., Kuzkin V.A. Derivation of Equations of State for Ideal Crystals of Simple Structure // Mech. Solids. 46 (3), 387-399 (2011) (Download pdf: 529 Kb)

Mie-Gruneisen equation of state[править]

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[править]

[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[править]

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[править]

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[править]

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[править]

  • Krivtsov A.M., Kuzkin V.A. Derivation of Equations of State for Ideal Crystals of Simple Structure // Mech. Solids. 46 (3), 387-399 (2011) (Download pdf: 529 Kb)
  • 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

Links[править]