Equations of State

Equations of State

In order to calculate the volume derivatives of the energy, an analytical expression needs to be fitted to the input \(E(V)\) data. This mathematical expression, commonly known as the equation of state (EOS), can take many forms. Which equation of state is used for a particular phase is determined by the FIT option to PHASE:

PHASE ... [FIT {POLYGIBBS|BM2|BM3|BM4|PT2|PT3|PT4|PT5|MURN|ANTONS|VINET|AP2|
                STRAIN {EULERIAN|BM|NATURAL|PT|LAGRANGIAN|LAGR|INFINITESIMAL|
                        INF|QUOTIENT|X1|X3|XINV3|X3INV|V} [order.i|0]}]
          [REG {LAD|LSQ}] [FIX i1.i v1.r i2.i v2.r ...]

Two types of EOS can be used: those based on strain polynomials and analytical expressions with a few parameters derived from the known elastic behavior of solids. The latter are very common in the literature and we refer to them as traditional EOS. The EOS is chosen using the FIT option to the PHASE keyword.

The analytical EOS is used in two distinct contexts:

  • The EOS is used to fit the static energy-volume data to generate the static equation equation of state.

  • At a given temperature, the EOS is used to fit the Hemlholtz free energy vs. volume data. This is used to calculate the volume at that temperature and an arbitrary pressure \(V(p,T)\) as well as the volume derivatives necessary for the calculation of the other thermodynamic properties.

Regardless of which equation of state is used to fit the data, it is strongly recommended that the quality of the fit is examined for every new set of data by at least plotting the efit file. This file presents the original source data and the fitted equation of state, and allows assessing whether the fit is good or not. Ideally, you would also examine the smoothness of some of the other thermodynamic properties that depend on the derivatives of the E(V) curve, such as the bulk modulus.

It is also important to note that gibbs2 is not designed for extrapolation. Thermodynamic properties are never calculated at equilibrium volumes outside of the input volume grid.

EOS Based on Strain Polynomials

The EOS based on strain polynomials are fitted using a linear least-squares fitting method (employing the SLATEC library). They are selected by using the FIT option to PHASE followed by the STRAIN keyword, the type of strain used, and the degree of the polynomial. The strain type can be one of:

Keyword Name Expression
EULERIAN or BM Birch-Murnaghan (Eulerian) \(f = \frac{1}{2}\left[\left(\frac{V}{V_0}\right)^{-2/3}-1\right]\)
NATURAL or PT Poirier-Tarantola (natural) \(f = \frac{1}{3}\log\left(\frac{V}{V_0}\right)\)
LAGRANGIAN Lagrangian \(f = \frac{1}{2}\left[\left(\frac{V}{V_0}\right)^{2/3}-1\right]\)
INFINITESIMAL Infinitesimal \(f = -\left(\frac{V}{V_0}\right)^{-1/3}+1\)
QUOTIENT or X1 Compression factor \(f = \frac{V}{V_0}\)
X3 Linear compression factor \(f = \left(\frac{V}{V_0}\right)^{1/3}\)
XINV3 or X3INV Inverse linear compression factor \(f = \left(\frac{V}{V_0}\right)^{-1/3}\)
V Volume \(f = V\)

The type of strain must be followed by the degree of the polynomial (order.i). For instance, FIT STRAIN BM 4 indicates a fourth-order Birch-Murnaghan EOS, equivalent to a fourth degree polynomial in the BM strain.

If 0 is used as the degree of the polynomial, a number of polynomials of increasing degree are fitted to the data and then a polynomial average is used. The agreement between polynomials of various degrees can be used as a measure of the quality of the input data. The minimum and maximum degree of the polynomials that enter the polynomial average are controlled by the SET MPAR and SET MPARMIN keywords. By default, the polymials go from third degree up to 12th degree. In cases when the input dataset contains only a few points, the default maximum strain polynomial degree is lower to prevent overfitting.

The default EOS in gibbs2 if no FIT keyword is given is to use an average of strain polynomials based on the Birch-Murnaghan strain (FIT STRAIN BM 0). This method has proved to be quite robust.

Traditional EOS

Traditional EOS based on analytical expressions can be used in gibbs2 as well. They are activated for a given phase by using the FIT option to the PHASE keyword followed by the keyword for the desired EOS. Contrary to the strain polynomials, these EOS are fitted using a non-linear minimization method (Levenberg-Marquardt) implemented in the MINPACK library. The goodness of the fit using traditional EOS is typically lower than with strain polynomials (particularly high-degree polynomials or averages) but these EOS contain fewer parameters so they are appropriate in cases when there are fewer points to fit, and they can also be used to extrapolate or to smooth noisy data.

The list of traditional EOS implemented and the corresponding keywords are:

Keyword Name Order Parameters Expression
BM2 Birch-Murnaghan 2 \(E_0,V_0,B_0\) \(E(V) = E_0 + \frac{9}{2} B_0 V_0 f^2\), \(f = \frac{1}{2} \left[ \left(\frac{V_0}{V}\right)^{2/3} - 1 \right]\)
BM3 Birch-Murnaghan 3 \(E_0,V_0,B_0,B_0'\) \(E(V) = E_0 + \frac{9}{2} V_0 B_0 f^2 [1 + (B_0^\prime-4) f]\), \(f\) as above
BM4 Birch-Murnaghan 4 \(E_0,V_0,B_0,B_0',B_0''\) \(E(V) = E_0 + \frac{3}{8} V_0 B_0 f^2 \{(9H - 63B_0^\prime + 143) f^2 + 12 (B_0^\prime-4) f + 12\}\), \(H = B_0 B_0^{\prime\prime} + (B_0^\prime)^2\), \(f\) as above
PT2 Poirier-Tarantola 2 \(E_0,V_0,B_0\) \(E(V) = E_0 + \frac{9}{2} B_0 V_0 f_N^2\), \(f_N = \ln\left(\frac{V}{V_0}\right)^{1/3}\)
PT3 Poirier-Tarantola 3 \(E_0,V_0,B_0,B_0'\) \(E(V) = E_0 + \frac{9}{2} B_0 V_0 f_N^2 [(B_0^\prime+2)f_N+1]\), \(f_N\) as above
PT4 Poirier-Tarantola 4 \(E_0,V_0,B_0,B_0',B_0''\) \(E(V) = E_0 + 9B_0 V_0 f_N^2 \{3(H+3B_0^\prime+3) f_N^2 + 4(B_0^\prime+2) f_N + 4\}\), \(H = B_0^{\prime\prime}B_0+(B_0^\prime)^2\), \(f_N\) as above
PT5 Poirier-Tarantola 5 \(E_0,V_0,B_0,B_0',B_0'',B_0'''\) Cetacean needed.
MURN Murnaghan 3 \(E_0,V_0,B_0,B_0'\) \(E(V) = E_0 + \frac{B_0 V}{B_0^\prime} \left[ \frac{(V_0/V)^{B_0^\prime}}{B_0^\prime -1} + 1 \right] - \frac{B_0 V_0}{B_0^\prime - 1}\)
ANTONS Anton-Schmidt 3 \(E_{\infty},V_0,B_0,B_0'\) \(E(V) = E_\infty + \frac{B_0 V_0}{n+1} \left(\frac{V}{V_0}\right)^{n+1} \left[\ln\left(\frac{V}{V_0}\right) - \frac{1}{n+1} \right]\), \(n = -B_0'/2\)
VINET Vinet 3 \(E_0,V_0,B_0,B_0'\) \(E(V) = E_0 + \frac{4B_0V_0}{(B_0^\prime-1)^2} - \frac{2B_0V_0}{(B_0^\prime-1)^2} [3(B_0^\prime-1)(\eta-1)+2] \exp\left\{-\frac{3}{2}(B_0^\prime-1)(\eta-1)\right\}\), \(\eta = (V/V_0)^{1/3}\)
AP2 Holzapfel’s AP2 3 \(E_0,V_0,B_0,B_0'\) \(E(V) = E_0 + 9B_0V_0 \Big\{\left[\Gamma(-2,c_0\eta)-\Gamma(-2,c_0)\right] c_0^2 e^{c_0} + \left[\Gamma(-1,c_0\eta)-\Gamma(-1,c_0)\right] c_0(c_2-1) e^{c_0} - \left[\Gamma( 0,c_0\eta)-\Gamma( 0,c_0)\right] 2c_2 e^{c_0} + \frac{c_2}{c_0} \left[e^{c_0(1-\eta)}-1\right] \Big\}\)

Note that the AP2 EOS requires setting the number of electrons for the system with the NELECTRON keyword. Also, the BM2, BM3, etc. EOS are different from the equivalent strain polynomial versions (STRAIN BM 2, STRAIN BM 3, etc.) only in that a non-linear least-squares fit is used. The use of the strain polynomial versions is recommended, as the two of them give equivalent results when the fit is successful and non-linear fits may fail sometimes.

Lastly, the fitting method used in the previous version of the program (gibbs) can be used with the POLYGIBBS keyword. This option is available mostly for testing and backwards-compatibility. POLYGIBBS may remove points from the energy-volume grid, and is known to be unstable in some cases.

Optional PHASE options

The following optional keywords can be used in PHASE to control how gibbs2 carries out the equation of state fitting for a particular phase:

REG {LAD|LSQ}

The REG keyword chooseS the regression technique for the EOS fits: least-squares (LSQ, default) or least absolute deviation (LAD). The former minmizes \(\sum_i |y_i-f(x_i)|^2\) and the latter minimizes \(\sum_i |y_i-f(x_i)|\). LAD is sometimes used as a robust fitting technique because it is less sensitive than least-squares to noise in the data. LAD can only be used with the traditional EOS, i.e., it cannot be used with FIT STRAIN X Y or with FIT POLYGIBBS.

FIX i1.i v1.r i2.i v2.r ...

The FIX keyword fixes some of the EOS parameters to user-defined values. This keyword only applies to traditional EOS (i.e. those that are not accessed via the FIT STRAIN keyword or POLYGIBBS). The constraints also apply to fits to static data only. They are not honored in the calculation of temperature-pressure data, i.e. when fitting free energy vs. volume curves. The i1.i integer gives the parameter to be fixed. It can be \(V_0\) (2), \(B_0\) (3), \(B_0'\) (4), \(B_0''\) (5), or \(B_0'''\) (6). The value at which the corresponding parameter is fixed comes after the integer identifier (v1.r). Multiple identifier/value pairs can be given.

Optional global keywords

The following optional keywords can be used in the gibbs2 input to control the EOS fitting for all phases:

NELECTRONS nelec.i

The NELECTRONS keyword gives the total number of electrons per NAT atoms. This value is used only if the AP2 EOS is used. Note this variable is not associated with the NELEC option in PHASE.

SET MPAR mpar.i

In the case of a strain polynomial EOS (FIT STRAIN) using weighed average polynomials (order.i = 0), MPAR sets the maximum degree of the weighed polynomial fit to mpar.i. The default mpar.i is 12 or half the E(V) points minus one, whichever is lowest. mpar.i is never lower than 3.

SET MPARMIN mparmin.i

In the case of a strain polynomial EOS using weighed average polynomials (order.i = 0), MPARMIN sets the minimum degree of the weighed polynomial fit to mparmin.i. The default mparmin.i is 2.

SET NDEL ndel.i

Sets the number of external points to be removed from the dataset in POLYGIBBS. POLYGIBBS should only be used for testing. The default ndel.i is 3.

SET PFIT_MODE {GAUSS|SLATEC}

With the GAUSS keyword, use the method from the previous gibbs program to conduct the linear least-squares fits (i.e. the strain polynomials and POLYGIBBS). SLATEC: use dpolft and dpolcf from the SLATEC library instead. GAUSS Is known to be unstable, so the default is SLATEC. This keyword is useful only for testing purposes.

SET PWEIGH_MODE {GIBBS1|GIBBS2|SLATEC}

Use the previous gibbs version (GIBBS1) or the GIBBS2 method to average the fit polynomials. Alternatively, let the SLATEC library do it. This keyword is useful only for testing purposes. Default: GIBBS2.