EOS Models

The mathematical descriptions of these models are presented below while the details of using them is presented in the description of the EOS API.

EOS Primer

An equation of state (EOS) is a constituitive model that generally relates thermodynamic quantities such as pressure, temperature, density, internal energy, free energy, and entropy consistent with the constraints of equillibrium thermodynamics.

singularity-eos contains a number of equations of state that are most useful in hydrodynamics codes. All of the equations of state presented here are considered “complete” equations of state in the sense that they contain all the information needed to derive a given thermodynamic quantity. However, not all variables are exposed since the emphasis is on only those that are needed for hydrodynamics codes. An incomplete equation of state is often sufficient for pure hydrodynamics since it can relate pressure to a given energy-density state, but it is missing information that relates the energy to temperature (which in turn provides a means to calculate entropy).

An excellent resource on equations of state in the context of hydrodynamics is the seminal work from Menikoff and Plohr. In particular, Appendix A contains a number of thermodynamic relationships that can be useful for computing additional quantities from those output in singularity-eos.

The Mie-Gruneisen form

Many of the following equations of state are considered to be of the “Mie-Gruneisen form”, which has many implications but for our purposes means that the general form of the EOS is

\[P - P_\mathrm{ref} = \rho \Gamma(\rho) (e - e_\mathrm{ref})\]

where ‘ref’ denotes quantities along some reference curve, \(P\) is the pressure, \(\rho\) is the density, \(\Gamma\) is the Gruneisen parameter, and \(e\) is the specific internal energy. In this sense, an EOS of this form uses the Gruneisen parameter to describe the pressure behavior of the EOS away from the reference curve. Coupled with a relationship between energy and temperature (sometimes as simple as a constant heat capacity), the complete equation of state can be constructed.

To some degree it is the complexity of the reference state and the heat capacity that will determine an EOS’s ability to capture the complex behavior of a material. At the simplest level, the ideal gas EOS uses a reference state at zero pressure and energy, while more complex equations of state such as the Davis EOS use the material’s isentrope. In ths way, the reference curve indicates the conditions under which you can expect the EOS to represent the intended behavior.

Nomenclature

The EOS models in singularity-eos are defined for the following sets of dependent and independent variables through various member functions described in the EOS API.

Function	Dependent Variable	Independent Variables
\(T(\rho, e)\)	Temperature	Density, Internal Energy
\(P(\rho, e)\)	Pressure	Density, Internal Energy
\(e(\rho, T)\)	Internal Energy	Density, Temperature
\(P(\rho, T)\)	Pressure	Density, Temperature
\(\rho(P, T)\)	Density	Pressure, Temperature
\(e(P, T)\)	Internal Energy	Pressure, Temperature
\(C_V(\rho, T)\)	Constant Volume Specific Heat Capacity	Density, Temperature
\(C_V(\rho, e)\)	Constant Volume Specific Heat Capacity	Density, Internal Energy
\(B_S(\rho, T)\)	Isentropic Bulk Modulus	Density, Temperature
\(B_S(\rho, e)\)	Isentropic Bulk Modulus	Density, Internal Energy
\(\Gamma(\rho, T)\)	Gruneisen Parameter	Density, Temperature
\(\Gamma(\rho, e)\)	Gruneisen Parameter	Density, Internal Energy

A point of note is that “specific” implies that the quantity is intensive on a per unit mass basis. It should be assumed that the internal energy is always specific since we are working in terms of density (the inverse of specific volume).

Disambiguation

Gruneisen Parameter

In this description of the EOS models, we use \(\Gamma\) to represent the Gruneisen coeficient since this is the most commonly-used symbol in the context of Mie-Gruneisen equations of state. This should be differentiated from

\[\gamma := \frac{C_P}{C_V} = \frac{B_S}{B_T}\]

though, which is the adiabatic exponent. Here \(C_P\) is the specific heat capacity at constant pressure and \(B_T\) is the isothermal bulk modulus.

Units and conversions

The default units for singularity-eos are cgs which results in the following units for thermodynamic quantities:

Quantity	Default Units	cgµ conversion	mm-mg-µs conversion
\(P\)	µbar	10^-12 Mbar	10^-10 GPa
\(\rho\)	g/cm³	1	1 mg/mm³
\(e\)	erg/g	10^-12 Terg/g	10^-10 J/mg
\(T\)	K	1	1
\(C_V\)	erg/g/K	10^-12 Terg/g/K	10^-10 J/mg/K
\(B_S\)	µbar	10^-12 Mbar	10^-10 GPa
\(\Gamma\)	unitless	–	–

Note: some temperatures are measured in eV for which the conversion is 8.617333262e-05 eV/K.

Sesame units are equivalent to the mm-mg-µs unit system.

Implemented EOS models

Ideal Gas

The ideal gas (aka perfect or gamma-law gas) model in singularity-eos takes the form

\[P = \Gamma \rho e\]

\[e = C_V T,\]

where quantities are defined in the nomenclature section.

The settable parameters are the Gruneisen parameter and specific heat capacity.

Although this differs from the traditional representation of the ideal gas law as \(P\tilde{V} = RT\), the forms are equivalent by recognizing that \(\Gamma = \frac{R}{\tilde{C_V}}\) where \(R\) is the ideal gas constant in units of energy per mole per Kelvin and \(\tilde{C_\mathrm{V}}\) is the molar heat capacity, relatable to the specific heat capacity through the molecular weight of the gas. Since \(\tilde{C_\mathrm{V}} = \frac{5}{2} R\) for a diatomic ideal gas, the corresponding Gruneisen parameter should be 0.4.

A common point of confusion is \(\Gamma\) versus \(\gamma\) with the latter being the adiabatic exponent. For an ideal gas, they are related through

\[\Gamma = \gamma - 1\]

Gruneisen EOS

One of the most commonly-used EOS to represent solids is the Steinberg variation of the Mie-Gruneisen EOS, often just shortened to “Gruneisen” EOS. This EOS uses the Hugoniot as the reference curve and thus is extremly powerful because the basic shock response of a material can be modeled using minimal parameters.

The pressure follows the traditional Mie-Gruneisen form,

\[P(\rho, e) = P_H(\rho) + \rho\Gamma(\rho) \left(e - e_H(\rho) \right),\]

Here the subscript \(H\) is a reminder that the reference curve is a Hugoniot. Other quantities are defined in the nomenclature section.

The above is an incomplete equation of state because it only relates the pressure to the density and energy, the minimum required in a solution to the Euler equations. To complete the EOS and determine the temperature, a constant heat capacity is assumed so that

\[T(\rho, e) = \frac{e}{C_V} + T_0\]

The user should note that this implies that \(e=0\) at the reference temperature, \(T_0\). Given this simple relationship, the user should treat the temperature from this EOS as only a rough estimate.

The Grunesien parameter is given by

\[\begin{split}\Gamma(\rho) = \begin{cases} \Gamma_0 & \rho < \rho_0 \\ \Gamma_0 \frac{\rho_0}{\rho} + b(1 - \frac{\rho_0}{\rho}) & \rho >= \rho_0 \end{cases}\end{split}\]

and when the unitless user parameter \(b=0\), this ensures the the Gruneisen parameter is of a form where \(\rho\Gamma =\) constant in compression.

The reference pressure along the Hugoniot is determined by

\[\begin{split}P_H(\rho) = \rho_0 c_0^2 \mu \begin{cases} 1 & \rho < \rho_0 \\ \frac{1 + \left(1 - \frac{1}{2}\Gamma_0 \right)\mu - \frac{b}{2} \mu^2} {\left(1 - (s_1 - 1)\mu + s_2 \frac{\mu^2}{1 + \mu} - s_3 \frac{\mu^3}{(1+\mu)^2} \right)^2} & \rho > \rho_0 \end{cases}\end{split}\]

where \(c_0\), \(s_1\), \(s_2\), and \(s_3\) are fitting paramters. The units of \(c_0\) are velocity while the rest are unitless.

Note that similar implementations may have subtly different definitions for the \(s_i\) coefficients and care must be taken to use the correct values. Since \(s_2\) and s_3 are unitless for example, these parameters would not directly correspond to coefficients in a polynomial fit of the shock velocity to the particle velocity.

JWL EOS

The Jones-Wilkins-Lee (JWL) EOS is used mainly for detonation products of high explosives. Similar to the other EOS here, the JWL EOS can be written in a Mie-Gruneisen form as

\[P(\rho, e) = P_S(\rho) + \rho w (e - e_S(\rho))\]

where the reference curve is an isentrope of the form

\[P_S(\rho) = A \mathrm{e}^{-R_1 \eta} + B \mathrm{e}^{-R_2 \eta}\]

\[e_S(\rho) = \frac{A}{\rho_0 R_1} \mathrm{e}^{-R_1 \eta} + \frac{B}{\rho_0 R_2} \mathrm{e}^{-R_2 \eta}.\]

Here \(\eta = \frac{\rho_0}{\rho}\) and \(R_1\), \(R_2\), \(A\), \(B\), and \(w\) are constants particular to the material. Note that the parameter \(w\) is simply the Gruneisen parameter and is assumed constant for the EOS (which is fairly reasonable since the detonation products are gasses).

Finally, to complete the EOS the energy is related to the temperature by

\[e = e_S(\rho) + C_V T\]

where \(C_V\) is the constant volume specific heat capacity.

Davis EOS

The Davis reactants and products EOS are both of Mie-Gruneisen forms that use isentropes for the reference curves. The equations of state are typically used to represent high explosives and their detonation products and the reference curves are calibrated to several sets of experimental data.

For both the reactants and products EOS, the pressure and energy take the forms

\[P(\rho, e) = P_S(\rho) + \rho\Gamma(\rho) \left(e - e_S(\rho) \right)\]

\[e(\rho, P) = e_S(\rho) + \frac{1}{\rho \Gamma(\rho)} \left(P - P_S(\rho) \right),\]

where the subscript \(S\) denotes quantities along the reference isentrope and other quantities are defined in the nomenclature section.

Davis Reactants EOS

The Davis reactants EOS uses an isentrope passing through a reference state and assumes that the heat capacity varies linearly with entropy such that

\[C_V = C_{V,0} + \alpha(S - S_0),\]

where subscript \(0\) refers to the reference state and \(\alpha\) is a dimensionless constant specified by the user.

The \(e(\rho, P)\) lookup is quite awkward, so the energy is more-conveniently cast in terms of termperature such that

\[e(\rho, T) = e_S(\rho) + \frac{C_{V,0} T_S(\rho)}{1 + \alpha} \left( \left(\frac{T}{T_S(\rho)} \right)^{1 + \alpha} - 1 \right),\]

which can easily be inverted to find \(T(\rho, e)\).

The Gruneisen parameter takes on a linear form such that

\[\begin{split}\Gamma(\rho) = \Gamma_0 + \begin{cases} 0 & \rho < \rho_0 \\ Zy & \rho >= \rho_0 \end{cases}\end{split}\]

where \(Z\) and \(y\) are dimensionless parameters.

Finally, the pressure, energy, and temperature along the isentrope are given by

\[\begin{split}P_S(\rho) = P_0 + \frac{\rho_0 A^2}{4B} \begin{cases} \mathrm{e} \left( 4By \right) -1 & \rho < \rho_0 \\ \sum\limits_{j=1}^3 \frac{(4By)^j}{j!} + C\frac{(4By)^4}{4!} + \frac{y^2}{(1-y)^4} & \rho >= \rho0 \end{cases}\end{split}\]

\[e_S(\rho) = e_0 + \int\limits_{\rho_0}^{\rho} \frac{P_S(\bar{\rho})}{\bar{\rho^2}}~\mathrm{d}\bar{\rho}\]

\[\begin{split}T_S(\rho) = T_0 \begin{cases} \left(\frac{\rho}{\rho_0} \right)^{\Gamma_0} & \rho < \rho_0 \\ \mathrm{e} \left( -Zy \right) \left(\frac{\rho}{\rho_0} \right)^{\Gamma_0 + Z} & \rho >= \rho_0 \end{cases}\end{split}\]

where \(A\), \(B\), \(C\), \(y\), and \(Z\) are all user-settable parameters and again quantities with a subcript of \(0\) refer to the reference state. The variable \(\bar{\rho}\) is simply an integration variable. The parameter \(C\) is especially useful for ensuring that the high-pressure portion of the shock Hugoniot does not cross that of the products.

The settable parameters are the dimensionless parameters listed above as well as the pressure, density, temperature, energy, Gruneisen parameter, and constant volume specific heat capacity at the reference state.

Davis Products EOS

The Davis products EOS is created from the reference isentrope passing through the CJ state of the high explosive along with a constant heat capacity. The constant heat capacity leads to the energy being a simple funciton of the temperature deviation from the reference isentrope such that

\[e(\rho, T) = e_S(\rho) + C_{V,0} (T - T_S(\rho)).\]

The Gruneisen parameter is given by

\[\Gamma(\rho) = k - 1 + (1-b) F(\rho)\]

where \(b\) is a user-settable dimensionless parameter and \(F(\rho)\) is given by

\[F(\rho) = \frac{2a (\rho V_{\mathrm{C}})^n}{(\rho V_{\mathrm{C}})^{-n} + (\rho V_{\mathrm{C}})^n}.\]

Here the calibration parameters \(a\) and \(n\) are dimensionless while \(V_{\mathrm{C}}\) is given in units of specific volume.

Finally, the pressure, energy, and temperature along the isentrope are given by

\[P_S(\rho) = P_{\mathrm{C}} G(\rho) \frac{k - 1 + F(\rho)}{k - 1 + a}\]

\[e_S(\rho) = e_{\mathrm{C}} G(\rho) \frac{1}{\rho V_{\mathrm{C}}}\]

\[T_S(\rho) = T_{\mathrm{C}} G(\rho) \frac{1}{(\rho V_{\mathrm{C}})^{ba + 1}}\]

where

\[G(\rho) = \frac{ \left( \frac{1}{2}(\rho V_{\mathrm{C}})^{-n} + \frac{1}{2}(\rho V_{\mathrm{C}})^n \right)^{a/n}} {(\rho V_{\mathrm{C}})^{-(k+a)}}\]

and

\[e_{\mathrm{C}} = \frac{P_{\mathrm{C}} V_{\mathrm{C}}}{k - 1 + a}.\]

Here, there are four dimensionless parameters that are settable by the user, \(a\), \(b\), \(k\), and \(n\), while \(P_\mathrm{C}\), \(e_\mathrm{C}\), \(V_\mathrm{C}\) and \(T_\mathrm{C}\) are tuning parameters with units related to their non-subscripted counterparts.