Introduction

 

This page outlines the structure and format of the thermodynamic data file read by Perple_X. This file contains the basic data for chemically pure entities. Here pure is used to mean the entity has a fixed chemical composition. Normally the entities correspond to real phases, i.e., stoichiometric compounds or phases corresponding to the endmember compositions of a solution phase, but in some cases the entity may correspond to a hypothetical state of matter or a function that is used to constrain a chemical potential as a function of pressure and temperature.

 

HSC/SUPCRT Apparent Gibbs Energy Conventions

 

In applied thermodynamics, energies are invariably computed relative to an arbitrary reference condition. There is no particular convention built into Perple_X, rather the convention is defined by the thermodynamic data. While there are many possible conventions only two, referred to here as the HSC and SUPCRT conventions, are widely used. In both, the Gibbs energy at elevated pressure and temperature is 

 

G(T,P) = G0 - integral(S(T,Pr),T=Tr..T) + integral(V(T,P),P=Pr..P)

 

where S(T,Pr) and V(T,P) are the absolute specific entropy and volume, respectively, and the subscript r denotes the reference condition (invariably 298.15 K and 1 bar). In the SUPCRT convention G0 is the Gibbs energy of formation from the elements at the reference condition, Gf(Tr,Pr), whereas in the HSC convention 

 

G0 = Hf(Tr,Pr) - Tr·S(Tr,Pr).

 

where Hf(Tr,Pr) is the enthalpy of formation from the elements. Given that 

 

Gf(Tr,Pr) = Hf(Tr,Pr) - Tr·(S(Tr,Pr) - Selements(Tr,Pr))

 

where Selements(Tr,Pr) is the stoichiometrically weighted sum of entropies of the elements in their respective stable forms at the reference condition, the apparent Gibbs energies computed by the SUPCRT and HSC conventions differ by a constant

 

G0HSC(T,P) - G0SUPCRT(T,P) =  Tr·Selements(Tr,Pr).

 

Thus, while it is clear that thermodynamic data can easily be converted from one convention to the other, it is important not to simply cut and paste data from one thermodynamic data file to another unless the data have the same convention.

 

The SUPCRT (Helgeson et al. 1978) and DEW (Harrison & Sverjensky 2013) programs and most early Perple_X data files assume the SUPCRT convention. In contrast, CALPHAD, THERMOCALC (Holland & Powell 1990), HSC (Roine 2002), and recent Perple_X data files generally assume the HSC convention.

 

Commonly Used Thermodynamic files

 

The following annotated list summarizes the most commonly used thermodynamic data files. The sources for data files not listed here can usually be found in the comments within the header section of the data file. These files can be downloaded from the datafile directory. Full references are listed in the bibliography. Endmembers are generally identified by abbreviated names, in some data files the header section contains a comprehensive list of abbreviations and the corresponding names and formulae (e.g., hp02ver.dat), it is always possible decipher unknown names by locating the individual entries within the relevant file. In a few cases, indicated below, the data base phase notation has been summarized in separate files.

Comment, Numerical data, and Keyword format

 

Thermodynamic data files consist of a header section followed by the data for individual entities. The different sections of the data base are identified by left justified, case sensitive, keywords. Numerical data is unformatted but individual numbers cannot be longer than 14 characters. Comments are indicated by the "|" character, comments can be placed on the same line as data, but, in general, comments should not be placed on the same line as a keyword that indicates the beginning or end of a section or data entry. Comments may also be written free form at the end of the header section before the "end" keyword.


 

Header Section

 

The subsections of the header section (Figure 1) are listed sequentially below:

 

Title: the first non-blank, non-comment, line is the data base title. The title may be up to 80 characters long.

   | comments are indicated by the | character.

This is the uninformative title of the data base            

begin_standard_variables 
P(bar)      1.00   0.1E-3 | if you don't like a 
T(K)      298.15   0.1E-4 | variable name, change it
Y(CO2)      0.00   0.1E-6
mu(C1)      0.00   0.1E-2
mu(C2)      0.00   0.1E-2
end_standard_variables

tolerance  .1E-2  

begin_components
NA2O    61.9790
MGO     40.3040
AL2O3  101.9610  | a comment can also
SIO2    60.0840  | be added like this
K2O     94.1960
CAO     56.0770  | the component names can be changed
TIO2    79.8660  | to whatever you prefer, e.g., CaO;
FEO     71.8440  | BUT remember to change in the rest
ZRO2   123.2200  | of the file as well (i.e., in the 
O2      31.9990  | formulae of individual entries).
H2O     18.0150
CO2     44.0100
end_components

begin_special_components
H2O
CO2
end_special_components

begin_makes       
           | some make definition examples:

           | a biotite solution endmember
fbi = 1 east -1/2 cor 1/2 hem 
13.45e3 0 0
           | quartz-fayalite-magnetite O2 buffer
qfm = 2 mt 3 q -3 fa         
0 0 0 
           | a melt endmember with modified stoichiometry
sil8L = 8/5 silL 
0 0 0 
           | a dqf corrected endmember
ts_dqf = 1 ts 
10000. 0. 0. 

end_makes 

end
 

Figure 1. Thermodynamic data file header section.

 

 

Standard variables: this section specifies names, reference values, and finite difference increments for the standard variables of the data base. The beginning of this section is indicated by the "begin_standard_variables" keyword and it is terminated by the "end_standard_variables" keyword. The specification of each variable is on a single line and consists of a left justified name (<9 characters), a number representing the variables reference or default value, and its finite difference increment. The increment is used only in unconstrained minimization calculations for tracing univariant phase fields (see page 15, card 6, of the somewhat out-of-date documentation vdoc for more information on the increments stored in array DELT). The units for thermodynamic potentials are J, K, bar and mol, these units are determined by the value of the universal gas constant as assigned in the block data subprogram (file tlib.f).

 

Tolerance: specified by the keyword "tolerance" followed by a numeric value. This tolerance is used only for unconstrained minimization calculations. This tolerance (DTOL) is used to determine whether a phase is stable with respect to a divariant assemblage with the same bulk composition as the phase in question. The tolerance has units of J/mol. The tolerances used to locate univariant (UTOL) and invariant (PTOL) equilibria are linear functions of DTOL (see page 15, card 6, of the out-of-date documentation vdoc for more information on DTOL/UTOL/PTOL). 

 

Components: this section assigns the names and gram-formula weights of the chemical components for the the data base. The beginning of this section is indicated by the "begin_components" keyword and it is terminated by the "end_components" keyword. The definition of each component is on a single line and consists of a left justified name (<6 characters), and the gram-formula weight of the component. The Perple_X program CTRANSF can be used to transform a data base from one set of components to another, provided the final components are a linear combination of the initial components.

 

Special components: this optional section identifies a subset of the data base components as special components. The beginning of this section is indicated by the "begin_special_components" keyword and it is terminated by the "end_special_components" keyword.  Ordinarily special components are the chemical components of a phase whose chemical composition can be used as an independent phase diagram variable (i.e., a saturated phase).

 

Make definitions: this optional section specifies make definitions. The beginning of this section is indicated by the "begin_make_definitions" keyword and it is terminated by the "end_make_definitions" keyword. Make definitions define the Gibbs energy of thermodynamic entities (e.g., chemical potential buffers, endmembers, etc.) as a linear combination of the Gibbs energies of real entities in the data section thermodynamic file. The definition optionally includes a pressure-temperature dependent increment, sometimes referred to as a DQF correction, that is added to the Gibbs energy of the linear combination.

 

The format of a make definition consists of two lines (<240 characters):

name = num1 * name1 + num2 * name2 +....+ numj * namej       
dnum1 dnum2 dnum3

 

where name is the name of the entity to be created, numi is a number or fraction (i.e., two numbers separated by a '/') indicating the stoichiometric proportion of an entity namei that exists as a data entry in the second part of the data file. The Gibbs energy of the made entity is

 

Gname = num1·Gname1 + num2·Gname2 +....+ numj·Gnamej + Gdqf        [2]

 

and

 

Gdqf[J/mol] = dnum1 + T[K]·dnum2 + P[bar]·dnum3        [3]


Perple_X 6.7.3+ allows more flexible input format for the coefficients of equation [3], the format is described at FAQ.

 

Made entities are excluded from calculations in Perple_X if they are included in the excluded phase list in the problem definition file (e.g., "in"); however, real entities to the right of the equals sign in a make definition are not effected by the excluded phase list. For example, if qL (liquid silica) is excluded, the make definition
 

q8L = 4 qL


will function, but qL will not be considered as a possible phase in the Perple_X calculation.

 

End-of-Header: the end of the header section is indicated by the "end" keyword.


 

Data Entries

 

The first line of data entries (Figure 2) for thermodynamic entities consist of a left justified name (<9 characters) for the entity followed by the EoS keyword, an equals sign, and an integer (the EoS keyword value) indicating the equation of state to be used for the entity. The EoS keyword may take the following values:

1  - normal polynomials for V, α, CP (e.g., Helgeson et al. 1978; Berman 1988)
2  - normal polynomials fpr α, CP, KT; Murnaghan for V (e.g., Holland & Powell 1998)
3  - normal polynomial for Cp; Ghiorso (2003) polynomial on α, KT; Birch-Murnaghan on pressure
4  - normal polynomials for α, CP, KT; Birch-Murnaghan for V (e.g., Saxena & Fei 1992)
5  - Debye Mie-Gruneisen (e.g., Stixrude & Bukowinski 1993)
6  - Debye Mie-Gruneisen (e.g., Stixrude & Lithgow-Bertelloni 2005b)
7  - normal polynomials for α, CP; exponential polynomial for V (e.g., Haas et al. 1986; Gottschalk 1997)

8  - normal polynomial for Cp; Mie-Gruneisen Einstein oscillator for thermal pressure; modified Tait for V (Holland & Powell, 2010)

9  - normal polynomials for Cp, α, KT; Tait for V (Holland & Powell, 2010)

10 - ideal gas, normal polynomial for Cp

11 - Mie-Gruneisen Liquid (e.g., Stixrude et al. 2009)

12 - Brosh/CALPHAD format

13 - normal polynomial for Cp; Komabayashi & Omori (2006) polynomial on α, KT, K'; Murnaghan on pressure

14 - Brosh/CALPHAD format for two part interpolative models

15 - Anderson Density Extrapolation model for aqueous species (Anderson et al. 1991)

16 - revised HKF formaulation for aqueous species (Tanger & Helgeson 1988, Shock et al. 1992, Sverjensky et al. 2013)

>100 - special EoS

The name and EoS keyword line is followed by a line that specifies the molar composition of the entitity in terms of the data base components. The format of this composition

 

name1(num1)name2(num2)name3(num3) ....

 

 
an       EoS = 2 | anorthite <= this is a comment          
AL2O3(1)SIO2(2)CAO(1)
G0 = -4007795 S0 = 200 V0 = 10.079  
c1 = 371.6 c2 = .12615E-1 c3 = -4110200 c5 = -2038.4  
b1 = .238E-4 b5 = -.238E-3 b6 = 960100 b7 = -137.85 b8 = 4
m0 = 421000 m1 = 3.48 m2 = -43  
transition = 1 type = 1  t1 = 2300  t2 = 11  t3 = .5E-1  
end
 

Figure 2. A thermodynamic data file entry.

 

where namei is the case-sensitive name of a component and numi is its stoichiometric coefficient in the molar formula unit of the entity. A component can occur only once in the composition, the composition must be written on a single line and cannot be separated from the name/EoS line by blank lines, but it can be separated from the name/EoS line by lines containing the comment character. Entities without mass (e.g., an entity representing a balanced chemical reaction) are indicated by the word "null" on the composition line.

 

The remaining lines assign values to the keywords: G0, S0, V0, c1-c7, d1-d10, m1-m3, and transition. The keywords can be listed in any order, if a keyword is not listed its value defaults to zero. The thermodynamic significance of the keyword values is dependent on specific keyword values, the possibilities are described here in two categories: the standard formulation (EoS 1-4, 7-10) and Mie-Gruneisen formulations (EoS 5, 6, 11). The EoS keyword can also be used to indicate certain internal equations of state.

 

Individual entries are terminated by the "end" keyword.


Standard Formulation (EoS 1-4 and 7-9)

 

In the standard formulation, the G0, S0, and V0 are the reference state Gibbs energy (J/mol), entropy (J/K/mol), and volume (J/bar) at the reference pressure (Pr) and temperature (Tr) of the data base.


Isobaric Heat Capacity, c1-c7

 

The values of c1-c7 correspond to the coefficients of the polynomial

 

CP(T,Pr) = c1 + c2·T + c3/T2 + c4·T2 + c5/T1/2 + c6/T + c7/T3  + c8·T3       [4]

 

that describes the isobaric heat capacity (CP) at the reference pressure.


Volumetric functions, b1-b8

 

The values of b1-b8 characterize the pressure-temperature dependence of the volume (V).


V(T,P): EoS 1

 

Volume (V, J/bar) at pressure (P) and temperature (T) is

 

V(T,P) = V0 + b2·(T-Tr) + b(P-Pr) + b(P-Pr)2 + b(T-Tr)2    [5]


V(T,P): EoS 7

 

Volume at pressure and temperature is

 

V(T,P) = V0·exp[b3·(T-Tr) + b(P-Pr)]   [6]


Temperature dependent volumetric functions: EoS 2, 4

 

The isobaric expansivity (α) at the reference pressure is

 

α(T,Pr) = b1 + b2·T + b3/T + b4/T2 + b5/T1/2        [7]

 

that describes the isobaric expansivity (α) at the reference pressure. Volume at the reference pressure is

 

V(T,Pr) = V0·[1 + integral(α(T,Pr),T=Tr..T)]        [8]

 

if the perplex_option.dat keyword approx_alpha is T (the default), or

 

V(T,Pr) = V0·exp[integral(α(T,Pr),T=Tr..T)]        [9]

 

if approx_alpha is F.

 

The isothermal bulk modulus is

 

K(T,Pr) = b6 + b7·(T-Tr)         [10]

 

if the perplex_option.dat keyword Anderson-Gruneisen is F, or

 

K(T,Pr) =  b6·exp[-K'·integral(α(T,Pr),T=Tr..T)]         [11]

 

if Anderson-Gruneisen is T.


Temperature dependent volumetric functions: EoS 3 

 

Volume at the reference pressure is computed as

 

V(T,Pr) = V0 + b1·(T-1673)     [12]

and the isothermal bulk modulus is

 

K(T,Pr) = V(T,Pr)/[b2 + b(T-1673)]         [13]


Temperature dependent volumetric functions and V(T,P): EoS 13

 

The isobaric expansivity (α) at the reference pressure is as in equation [7]. For consistency with Komabayashi & Omori (PEPI 156:89-107, 2006) approx_alpha and Anderson-Gruneisen are ignored.

 

The isobaric expansivity (α) at the reference pressure is

 

α(T,Pr) = b1 + b2·T + b3/T + b4/T2         [34]

 

 

The bulk modulus at the reference pressure is  

 

K(T,Pr) = 1/(b5 + b6·T + b7·T2 + b8·T3)       [35]

 

and its pressure derivative is 

 

K' = b9 + b10·(T-Tr)· ln(T/Tr).        [36]

 

V(T,Pr) is computed as in equation [9] and V(T,P) is computed as in equation [14] below.


V(T,P): EoS 2, 3, 4

 

Perple_X uses the Murnaghan or Birch-Murnaghan isothermal equations of state (third order) to compute volume depending on the value of b8. If b8 > 0, b8 is the pressure derivative of the bulk modulus (K') and the Murnaghan equation is used 

 

V(T,P) = V(T,Pr)·[1 - K'·P/{K'·P + K(T,Pr)}]1/K'         [14]

 

is used. Otherwise if b8 < -3, b8 = -K' and the Birch-Murnaghan equation

 

P = 3·K(T,Pr)·f·[1 + f]5/2·[1 + {3·(K'-4)/4}·f],         [15]

 

where

 

f =[{V(T,Pr)/V(T,P)}2/3 - 1]/2,        [16]

 

is used and solved numerically for V(T,P).


V(T,P): EoS 8 and 9

 

Documentation pending publication of Holland & Powell (2010, JMG)

 

EoS 8

b1 - alpha0 (1/K)
b5 - Debye T (K)
b6 - K0 (J/mol)
b7 - K0" (1/bar)
b8 - K0'

EoS 9

b1 - alpha0 (1/K)
b5 - dK/dT (bar/K)
b6 - K0 (J/mol)
b7 - K0" (1/bar)
b8 - K0'

Adiabatic Shear Modulus, m0-m2

The adiabatic shear modulus (bar) is

μS(T,P) = m0 + m1·(P-Pr) + m2·(T-Tr)        [17]

if the perplex_option.dat keyword poisson_ratio is off or on; if m0-m2 are not specified and poisson_ratio is on, or if poisson_ratio is all, then

μS(T,P) = ν·KS(T,P)        [18]

where the Poisson ratio ν is specified by the second value of the poisson_ratio keyword and the adiabatic bulk modulus KS(T,P) is computed by differentiation of the Gibbs energy (Equation 4 of Connolly & Kerrick 2002).


Adiabatic Bulk Modulus, k0-k2

By default, Perple_X computes the adiabatic bulk modulus KS(T,P) by differentiation of the Gibbs energy (Equation 4 of Connolly & Kerrick 2002). This method has the virtue that the modulus is consistent with the equation of state used to compute phase relations. The importance of this consistency for seismic velocity is debatable because phase relations depend directly on the Gibbs energy, whereas KS(T,P) is dependent primarily on second order derivatives of the Gibbs energy. Thus it can be argued that in some cases empirically calibrated expressions for KS(T,P) are more accurate than the values obtained from an EoS intended only for phase equilibrium calculations (e.g., Holland & Powell 1998). To accommodate this case, the adiabatic bulk modulus (bar) can be specified explicitly in the thermodynamic data file by the expression

KS(T,P) = k0 + k1·(P-Pr) + k2·(T-Tr).        [18.1]

This explicit expression for the KS(T,P) is used only if the perplex_option.dat keyword explicit_bulk_modulus = T (true). If the coefficients are not specified or explicit_bulk_modulus = F, then KS(T,P) is computed by differentiation of the Gibbs energy.


Mie-Gruneisen Formulations for Solids (EoS 5-6)

The Mie-Gruneisen type formulations of Stixrude & Bukowinskii (1993) and Stixrude & Lithgow-Bertelloni (2005b) are specified in Perple_X by EoS values 5 and 6, respectively. Both formulations use the Debye thermal model for which the iscohoric Helmoltz free energy is

A(T,V) = (Θ/T)-3·integral((Θ/T)2·ln{1 - exp[-(Θ/T)]},(Θ/T))        [19]

and the third-order Birch-Murnaghan equation of state. The formulations differ from each other in the dependence of the Debye temperature (Θ) on volume. In the Stixrude & Bukowinskii (1993) formulation, the Debye temperature is

Θ = Θ0·exp{-γ0[{V/V0}q0 - 1]/q0}        [20]

and in the Stixrude & Lithgow-Bertelloni (2005b) formulation

Θ = Θ0·a2·[1 + (a1 + a2·f/2)·f]1/2        [21]

with

a1 = 6·γ0        [22]

a2 = a1·[a1 - 2 - 3q0]        [23]

and f as in equation 16.

For the Stixrude & Bukowinskii (1993, EoS=5) formulation the correspondence of the Perple_X keywords is [the zero subscript indicates a value at reference conditions]:

  • G0 = Helmoltz energy (F0, J/mol)
  • S0 = number of atoms per formula unit (n)
  • V0 = negative of the volume (-V0)
  • c1 = isothermal bulk modulus (K0, bar)
  • c2 = pressure derivative of the isothermal bulk modulus (K')
  • c3 = Debye Temperature (Θ0, K)
  • c4 = Gruneisen thermal parameter (γ0)
  • c5 = Mie-Gruneisen exponent (q0)
  • c6 = Shear strain derivative of the tensorial Gruneisen parameter (ηS0)
  • c7 = Configurational (and magnetic) entropy (J/mol/K)
  • m1 = Adiabatic shear modulus (μS0, bar)
  • m2 = Pressure derivative of the adiabatic shear modulus (μS0')
  • The only difference from the above list in the parameter assignments for the Stixrude & Lithgow-Bertelloni (2005b, EoS=6) formulation is

  • S0 = negative of the number of atoms per formula unit (-n)

  • Modified Anderson Density Model for Aqueous Species (EoS 15)

    EoS 15 implements the Anderson density model (Anderson et al. 1991) for estimation of the properties of aqueous species at elevated pressure and temperature as modified by Holland & Powell (1998, page 313). The tags used to identify the parameters of this model in Perple_X thermodynamic data files are: 

  • G0 = reference state Gibbs energy (J/mol)
  • S0 = reference state entropy (J/mol/K)
  • V0 = reference state the volume (J/bar)
  • c1 = reference state heat capacity (J/mol/K)
  • c2 = temperature dependence of the reference state heat capacity (J/mol/K2)
  • c3 = ionic charge
  • In addition to the above species specific parameters, the following solvent (water) properties (Anderson et al. 1991) are assumed for the evaluation of EoS 15: 

  • α0 = reference state expansivity = 25.93·10-5/K
  • β0 = reference state compressibility = 45.23·10-6/bar
  • (dα/dT)0 = reference state derivative of α with respect to T = 9.5714·10-6/K2
  • ρ = density (Pitzer & Sterner 1994).

  • Revised HKF Formulation for Aqueous Species (EoS 16)

    EoS 16 implements the revised HKF formulation (Tanger & Helgeson 1988, Shock et al. 1992) for the estimation of the thermodynamic properties of aqueous species at elevated pressure and temperature. The tags, listed below, used to identify the parameters of this model in Perple_X thermodynamic data files correspond to the notation used in Shock et al. (1992) and the Deep Earth Water (DEW) Model (Harrison & Sverjensky, 2013).

  • G0 = reference state Gibbs energy (J/mol)
  • S0 = reference state entropy (J/mol/K)
  • ω0 = reference state Born coefficient (J/mol)
  • c1 = caloric equation of state parameter (J/mol/K)
  • c2 = caloric equation of state parameter (J-K/mol)
  • a1 = volumetric equation of state parameter (J/mol/bar)
  • a2 = volumetric equation of state parameter (J/mol)
  • a3 = volumetric equation of state parameter (J-K/bar/mol)
  • a4 = volumetric equation of state parameter (J-K/mol)
  • In addition to the above species specific parameters, the following solvent (water) properties/functions are assumed for the evaluation of EoS 16: 
  • θ = characteristic temperature, 228 K
  • ψ = characteristic pressure, 2600 bar
  • ε = dielectric constant (Sverjensky et al. 2013)
  • Y0 = (dlnε/dT)/ε [at Tr (298.15 K) and Pr (1 bar)] = -5.79865·10-5/K
  • ρ = density of water (Zhang & Duan 2005)
  • η = proportionality constant relating the Born constant to the inverse of electrostatic radius = 4.184·1.66027·105 J-Angstrom/mol

  • Special EoS (EoS > 100)

    Perple_X has a number of specialized internal EoS, a resume of these follows


    Explicit Thermal Disorder, d1-d9

     

    Coefficients d1-d9 provide for an explicit expression of the energetic effects of thermal disordering (Berman & Brown 1984; Berman 1988) in a thermodynamic entity. The values d1-d7 correspond to the coefficients of the polynomial

     

    CP,dis = d1 + d2·T + d3/T2 + d4·T2 + d5/T1/2 + d6/T + d7/T2        [24]

     

    which represents the disorder component of the isobaric heat capacity. This component is added to the baseline heat capacity over the temperature(K) interval d8 to d9.


    Lambda Transitions, transition, type, t1-t11

    The data for a Lambda (and mock-Lambda) transition of a thermodynamic entity is entered on a single line with the general format

    transition = num1 type = num2 t1 = num3 ... t11 = num13

    transition and type are mandatory keywords and must be the first and second entries for the transition, the remaining keywords are optional. The transition keyword value is the index of the transition; normally 1, but for phases with more than one transition the index indicates the transition's order of occurrence with respect to temperature at the reference pressure. The type keyword value indicates the model to be used to interpret the subsequent data keywords (t1-t12) and can take the following values:


    type = 1, UBC form

    In the UBC lambda model, the lambda component of the Gibbs energy is evaluated in terms of the variable T' = T + Td, where Td = t4·(P-Pr). The lambda component of the heat capacity is

    C = T'·(t1 + t2·T')        [25]

    over the interval from t7 - Td to the smaller of T' or t3 - Td. t5 and t6 approximate the ∂V/∂P and ∂V/∂T derivatives for the entity, and t8 is an energetic correction.


    type = 2, Standard form

    In the standard lambda model, transitions are approximated as discontinuous transitions and each phase is characterized by a unique isobaric heat capacity function. No correction is made for the effects of pressure on the transition temperature. The parameters are

    t1 - transition temperature at the reference pressure
    t3 - entropy change of the transition
    t4-t11 - coefficients c1-c8 of the heat capacity function to be used at T > t1

    type = 3, Helgeson form

    The Helgeson model is identical to the standard model except that it accounts for the pressure dependence of the transition by introducing the additional parameter

    t3 - Clapeyron slope (∂P/∂T) of the transition

    type = 4, Landau form

    The Landau model is based on three parameters: t1 - the temperature T0 at which the phase becomes fully disordered at the reference pressure; t2 - the maximum entropy change of disorder, Sd; t3 - the maximum volume change of disorder, Vd. The Gibbs energy change of ordering due to a lambda transition is

    ΔG = h' - T·s' + integral(v',P=Pr..P) + GLandau        [26]

    where

    h' = Sd·T0·(Q02 - Q06/3)        [27]

    s' = Sd·Q02        [28]

    v' = Vd·Q02·[1 - K'·P/{K'·P + K(T,Pr)}]1/K'·[1 + integral(α(T,Pr),T=Tr..T)]        [29]

    Tc = T0 + P·Vd/Sd        [30]

    Q0 = (1 - Tr/T0)1/4        [31]

    if T < Tc, then Q = (1 - T/T0)1/4, else Q = 0        [32]

    GLandau = Sd·[(T - TcQ2 + Tc·Q6/3)]        [33]

    and the unspecified properties are those of the completely disordered state.


    type = 5, Bragg-Williams form

    Documentation pending publication of Holland & Powell (2011, JMG)

    t1 - enthalpy change of disordering (J/mol)
    t2 - volume change of disordering (J/bar/mol)
    t3 - W_H (J/mol)
    t4 - W_V (J/bar/mol)
    t5 - n
    t6 - fac

    Bibliography

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