An alternative use of Kieffer’s lattice dynamics model using vibrational density of states for constructing thermodynamic databases

Abstract

We use Kieffer’s model to represent the vibrational density of states (VDoS) and thermodynamic properties of pure substances in pressure–temperature space. We show that this model can be simplified to a vibrational model in which the VDoS is represented by multiple Einstein frequencies without significant loss of accuracy in thermodynamic properties relative to experimental data. The resulting analytical expressions for thermodynamic properties, including the Gibbs energy, are mathematically simple and easily accommodated in existing computational software for making thermodynamic databases. We show for aluminium, platinum, orthoenstatite and forsterite that thermodynamic properties can be represented with comparable accuracy as with Kieffer’s model with the same number of fitting parameters as in the Mie–Grüneisen–Debye model. We demonstrate that the method enables to achieve thermodynamic properties with superior accuracy relative to the Mie–Grüneisen–Debye method. The method is versatile in the sense that it allows incorporating dispersion of Grüneisen parameters. It is possible to extend the formalism to include other physical effects, such as intrinsic anharmonicity in the same way as in vibrational models based on Kieffer’s formalism.

Appendix

Thermodynamic expressions of properties in which the VDoS is represented by multiple Einstein oscillators are derived from the Helmholtz energy. Vinet et al.’s (1987, 1989) equation of state for the static lattice has been used. For aluminium, the electronic contribution to the Helmholtz energy is described with a free electron gas model containing the electronic Grüneisen parameter γ _el and the electronic coefficient β _el. $V_{0}^{\text{st}}$, $K_{0}^{\text{st}}$ and $K_{0}^{{\prime {\text{st}}}}$ denote the volume, bulk modulus and its pressure derivative for the static lattice in which vibrational motions are absent. V ₀ represents the volume of the vibrating lattice at zero Kelvin and zero pressure. U ^ref is a reference energy at zero Kelvin and zero pressure and adjusted in such a way that the enthalpy at 298.15 K and 1 bar pressure is represented. For forsterite and orthoenstatite, we used values given by Robie and Hemingway (1995), whereas for elemental aluminium and platinum heat of formations at ambient conditions are zero. The expressions below are for the quasi-harmonic case and therefore do not contain an intrinsic anharmonicity contribution.

In the equations below, we define the volume ratio, η as:

$$ \eta = \frac{V}{{V_{0}^{\text{st}} }}.$$

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Helmholtz energy

$$ \begin{aligned} A(V,T) =\,& U^{\text{ref}} + \frac{{4K_{0}^{\text{st}} V_{0}^{\text{st}} }}{{\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)^{2} }}\left\{ {1 + \left[ {\frac{3}{2}\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)\left( {1 - \eta^{1/3} } \right) - 1} \right]\exp \left[ {\frac{3}{2}\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)\left( {1 - \eta^{1/3} } \right)} \right]} \right\} \\ & \quad + 3nRT\sum\limits_{j = 1}^{{N_{\text{E}} }} {f_{j} \left[ {\frac{{\theta_{j}^{\text{E}} }}{2T} + \ln \left( {1 - \exp \left( { - \frac{{\theta_{j}^{\text{E}} }}{T}} \right)} \right)} \right]} - \frac{{\beta_{\text{el}} }}{2}\left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T^{2} \\ \end{aligned} .$$

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Energy

$$ \begin{aligned} U(V,T) = & U^{\text{ref}} + \frac{{4K_{0}^{\text{st}} V_{0}^{\text{st}} }}{{\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)^{2} }}\left\{ {1 + \left[ {\frac{3}{2}\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)\left( {1 - \eta^{1/3} } \right) - 1} \right]\exp \left[ {\frac{3}{2}\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)\left( {1 - \eta^{1/3} } \right)} \right]} \right\} \\ & \quad + 3nR\sum\limits_{j = 1}^{{N_{\text{E}} }} {\left\{ {f_{j} \theta_{j}^{\text{E}} \left\{ {\frac{1}{2} + \frac{1}{{\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1}}} \right\}} \right\}} + \frac{{\beta_{\text{el}} }}{2}\left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T^{2} \\ \end{aligned}. $$

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Isochoric heat capacity

$$ C_{V} (V,T) = 3nR\sum\limits_{j = 1}^{{N_{\text{E}} }} {\left\{ {f_{j} \frac{{\left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)^{2} \exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)}}{{\left[ {\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1} \right]^{2} }}} \right\}} + \beta_{\text{el}} \left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T. $$

(16)

Entropy

$$ S(V,T) = 3nR\sum\limits_{j = 1}^{{N_{\text{E}} }} {\left\{ {f_{j} \left\{ { - \ln \left[ {1 - \exp \left( { - {\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)} \right] + \frac{{{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}}}{{\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1}}} \right\}} \right\}} + \beta_{\text{el}} \left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T. $$

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Isothermal bulk modulus

$$ \begin{aligned} K(V,T) = & K_{0}^{\text{st}} \left\{ {2\eta^{ - 2/3} - \eta^{ - 1/3} + \frac{3}{2}\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)\left( {\eta^{ - 1/3} - 1} \right)} \right\}\exp \left\{ {\frac{3}{2}\left( {K_{0}^{{\prime {\text{st}}}} - 1} \right)\left( {1 - \eta^{1/3} } \right)} \right\} \\ & \quad + \frac{3nRT}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\left\{ {f_{j} \left\{ {\left( {\frac{1}{2} + \frac{1}{{\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1}}} \right) \cdot \frac{{\theta_{j}^{\text{E}} }}{T} \cdot \left( {\gamma_{j}^{2} + \gamma_{j} - V\left( {\frac{{\partial \gamma_{j} }}{\partial V}} \right)_{T} } \right) - \frac{{\gamma_{j}^{2} \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)^{2} \exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)}}{{\left[ {\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1} \right]^{2} }}} \right\}} \right\}} \\ & \quad - \frac{{\beta_{\text{el}} \gamma_{\text{el}} (\gamma_{\text{el}} - 1)}}{2V}\left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T^{2} \\ \end{aligned}. $$

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Temperature derivative of pressure

$$ \left( {\frac{\partial P}{\partial T}} \right)_{V} = \left( {\frac{{\partial P^{\text{vib}} }}{\partial T}} \right)_{V} = \frac{3nR}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\left\{ {f_{j} \gamma_{j} \frac{{\left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)^{2} \exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right)}}{{\left[ {\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1} \right]^{2} }}} \right\}} + \frac{{\beta_{\text{el}} \gamma_{\text{el}} }}{V}\left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T. $$

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Thermal expansivity

Thermal expansivity is calculated from the total bulk modulus, Eq. (18) and temperature derivative of pressure, Eq. (19).

$$ \alpha = \frac{1}{K}\left( {\frac{\partial P}{\partial T}} \right)_{V}. $$

(20)

Isobaric heat capacity

Isobaric heat capacity is calculated from isochoric heat capacity, Eq. (16), bulk modulus, Eq. (18), and thermal expansivity Eq. (20).

$$ C_{P} = C_{V} + \alpha^{2} KVT. $$

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Total pressure

$$ \begin{aligned} P(V,T) = & 3K_{0}^{\text{st}} \left[ {\eta^{ - 2/3} - \eta^{ - 1/3} } \right]\exp \left\{ {\frac{3}{2}\left( {K_{0}^{\prime st} - 1} \right)\left[ {1 - \eta^{1/3} } \right]} \right\} \\ & \quad + \frac{3nR}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\left\{ {f_{j} \gamma_{j} \left\{ {\frac{{\theta_{j}^{\text{E}} }}{2} + \frac{{\theta_{j}^{\text{E}} }}{{\exp \left( {{\raise0.7ex\hbox{${\theta_{j}^{\text{E}} }$} \!\mathord{\left/ {\vphantom {{\theta_{j}^{\text{E}} } T}}\right.\kern-0pt} \!\lower0.7ex\hbox{$T$}}} \right) - 1}}} \right\}} \right\}} + \frac{{\beta_{\text{el}} \gamma_{\text{el}} }}{2V}\left( {\frac{V}{{V_{0} }}} \right)^{{\gamma_{\text{el}} }} T^{2} \\ \end{aligned}. $$

(22)

Gibbs energy

The Gibbs energy is calculated from the Helmholtz energy, Eq. (14) and pressure Eq. (22).

$$ G(P,T) = A(T,V) + P(V,T) \cdot V. $$

(23)

Enthalpy

The enthalpy is calculated from the energy, Eq. (15) and pressure, Eq. (22).

$$ H(P,T) = U(T,V) + P(V,T) \cdot V. $$

(24)

Equations (14)–(24) are used as follows. At a selected condition of pressure and temperature, and input parameters as given in, for example, Table 2, the volume is calculated from Eqs. (9), (10) and (22). The calculation of volume is achieved numerically in an iteration process. Once volume has been established, all thermodynamic properties are calculated analytically.

Contributions due to intrinsic anharmonicity

The expressions presented above can be extended to include intrinsic anharmonicity. The formalism presented by Jacobs and de Jong (2005) requires expressions in which the quasi-harmonic expressions above are quite complicated to use. Here, we present an alternative based on a thermodynamic perturbation formalism presented by Oganov and Dorogokupets (2004). In that case, the expressions below should be added to the corresponding expressions for thermodynamic properties. The anharmonicity parameter is given by:

$$ a_{j} = \left( {\frac{{\partial \ln \theta_{j}^{\text{E}} }}{\partial T}} \right)_{V}. $$

(25)

The first and second volume derivatives at constant temperature, necessary in thermal pressure, its isochoric temperature derivative and bulk modulus are given as $a_{j}^{\prime }$ and $a_{j}^{\prime \prime }$. The volume dependence of the anharmonicity parameter was used for platinum, but not for aluminium, forsterite and orthoenstatite. Its usual expression is

$$ a_{j} (V) = a_{j,0} \left( {\frac{V}{{V_{0} }}} \right)^{z}. $$

(26)

In Eq. (26), a _j,0 is the anharmonicity parameter at zero Kelvin, zero pressure and z a constant. To obtain shorter expressions below, we define

$$ x_{j} = \frac{{\theta_{j}^{\text{E}} }}{T}. $$

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Note that the Einstein temperature in the thermodynamic formulation for the Helmholtz energy of Oganov and Dorogokupets (2004) and the expressions below is given by the quasi-harmonic expressions, Eqs. (9, 10) and that not the temperature and volume dependent property resulting from Eq. (25) should be used.

Anharmonic contribution to entropy

$$ S_{\text{anh}}^{\text{vib}} = - 3nRT\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j} }}{6}f_{j} \left\{ {\frac{{3x_{j}^{3} e^{{x_{j} }} \left( {e^{{x_{j} }} + 1} \right)}}{{\left( {e^{{x_{j} }} - 1} \right)^{3} }}} \right\}}. $$

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Anharmonic contribution to isochoric heat capacity

$$ C_{{V,{\text{anh}}}}^{\text{vib}} = - 3nRT\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j} }}{6}f_{j} \left\{ {\frac{{3x_{j}^{3} e^{{3x_{j} }} (x_{j} - 2) + 12x_{j}^{4} e^{{2x_{j} }} + 3x_{j}^{3} e^{{x_{j} }} (2 + x_{j} )}}{{\left( {e^{{x_{j} }} - 1} \right)^{4} }}} \right\}}. $$

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Anharmonic contribution to thermal pressure

$$ \begin{aligned} P_{\text{anh}}^{\text{vib}} = & \frac{{3nRT^{2} }}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j} \gamma_{j} }}{6}f_{j} } \left\{ {\frac{{x_{j}^{2} }}{2} + \frac{{3x_{j}^{2} (2 - x_{j} )e^{{2x_{j} }} - 3x_{j}^{2} (2 + x_{j} )e^{{x_{j} }} }}{{\left( {e^{{x_{j} }} - 1} \right)^{3} }}} \right\} \\ & \quad - 3nRT^{2} \sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j}^{\prime } }}{6}f_{j} \left\{ {\frac{{x_{j}^{2} }}{4} + \frac{{3x_{j}^{2} e^{{x_{j} }} }}{{\left( {e^{{x_{j} }} - 1} \right)^{2} }}} \right\}} \\ \end{aligned}. $$

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Anharmonic contribution to isochoric temperature derivative of thermal pressure

$$ \begin{aligned} \left( {\frac{{\partial P_{\text{anh}}^{\text{vib}} }}{\partial T}} \right)_{V} = & \frac{3nRT}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j} \gamma_{j} }}{6}} f_{j} \left\{ {\frac{{3x_{j}^{3} e^{{3x_{j} }} (3 - x_{j} ) - 12x_{j}^{4} e^{{2x_{j} }} - 3x_{j}^{3} e^{{x_{j} }} (3 + x_{j} )}}{{\left( {e^{{x_{j} }} - 1} \right)^{4} }}} \right\} \\ & \quad - 3nRT\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j}^{\prime } }}{6}f_{j} } \left\{ {\frac{{3x_{j}^{3} e^{{x_{j} }} \left( {e^{{x_{j} }} + 1} \right)}}{{\left( {e^{{x_{j} }} - 1} \right)^{3} }}} \right\} \\ \end{aligned}. $$

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Anharmonic contribution to isothermal bulk modulus

$$ \begin{aligned} K_{\text{anh}}^{\text{vib}} = & \frac{{3nRT^{2} }}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j} }}{6}f_{j} \left[ {\gamma_{j} - V\left( {\frac{{\partial \gamma_{j} }}{\partial V}} \right)_{T} } \right]} \left[ {\frac{{x_{j}^{2} }}{2} + \frac{{3x_{j}^{2} (2 - x_{j} )e^{{2x_{j} }} - 3x_{j}^{2} (2 + x_{j} )e^{{x_{j} }} }}{{\left( {e^{{x_{j} }} - 1} \right)^{3} }}} \right] \\ & \quad + \frac{{3nRT^{2} }}{V}\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j} \gamma_{j}^{2} }}{6}f_{j} } \left[ {x_{j}^{2} + \frac{{e^{{3x_{j} }} (12x_{j}^{2} - 15x_{j}^{3} + 3x_{j}^{4} ) + e^{{2x_{j} }} ( - 24x_{j}^{2} + 12x_{j}^{4} ) + e^{{x_{j} }} (12x_{j}^{2} + 15x_{j}^{3} + 3x_{j}^{4} )}}{{\left( {e^{{x_{j} }} - 1} \right)^{4} }}} \right] \\ & \quad + 3nRT^{2} V\sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j}^{\prime \prime } }}{6}f_{j} \left[ {\frac{{x_{j}^{4} }}{4} + \frac{{3x_{j}^{2} e^{{x_{j} }} }}{{\left( {e^{{x_{j} }} - 1} \right)^{2} }}} \right]} \\ & \quad - 3nRT^{2} \sum\limits_{j = 1}^{{N_{\text{E}} }} {\frac{{a_{j}^{\prime } \gamma_{j} }}{6}} f_{j} \left[ {\frac{{x_{j}^{2} }}{2} + \frac{{3x_{j}^{2} (2 - x_{j} )e^{{2x_{j} }} - 3x_{j}^{2} (2 + x_{j} )e^{{x_{j} }} }}{{\left( {e^{{x_{j} }} - 1} \right)^{3} }}} \right] \\ \end{aligned}. $$

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A computer program written in Fortran for the calculation of thermodynamic properties of pure substances in pressure–temperature space will be made available on request.