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.
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,P_{r}),T=T_{r}..T) + integral(V(T,P),P=P_{r}..P)
where S(T,P_{r}) 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, G_{f}(T_{r},P_{r}), whereas in the HSC convention
G0 = H_{f}(T_{r},P_{r}) - T_{r}·S(T_{r},P_{r}).
where H_{f}(T_{r},P_{r}) is the enthalpy of formation from the elements. Given that
G_{f}(T_{r},P_{r}) = H_{f}(T_{r},P_{r}) - T_{r}·(S(T_{r},P_{r}) - S^{elements}(T_{r},P_{r}))
where S^{elements}(T_{r},P_{r}) 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
G0^{HSC}(T,P) - G0^{SUPCRT}(T,P) = T_{r}·S^{elements}(T_{r},P_{r}).
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.
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.
b89ver.dat, b92ver.dat - Berman (1988) data base and update. SUPCRT convention.
bb84ver.dat - Berman & Brown (1984), CaO-Al2O3-SiO2 liquidus data base. SUPCRT convention.
ba96ver.dat - Berman & Aranovich (1996) data base. SUPCRT convention.
cr_hp02ver.dat - a modification of the hp02ver.dat data base that includes some Cr endmembers. The modifications are documented by Klemme et al. (2009, updated by Ziberna et al. 2013). SUPCRT convention.
DEW13ver.dat - Harrison & Sverjensky (2013) aqueous species data base for the Deep Earth Water (DEW) Model (deepcarbon.net/feature/deep-earth-water-model-download-now#.V1WiwPl965M). This compilation is an extension and revision of the SUPCRT data base for aqueous species. The DEW spreadsheet data can be converted to Perple_X format using the Perple_X program DEW_2_ver, which allows the user to choose the elements of interest and apparent Gibbs energy convention, the on-line version of this file assumes the SUPCRT convention.
g97ver.dat - Gottschalk (1997). SUPCRT convention.
hp02ver.dat - The Holland & Powell (1998) data base as revised by the authors in 2002, database notation. This data base provides the most comprehensive chemical model for silicates in the lithosphere. The hp02ver.dat file is appropriate for calculations at depths < 440 km. The Holland & Powell database has been augmented by shear moduli for seismic velocity calculations. The references for these moduli are summarized in Connolly & Kerrick (2002) and Connolly (2005). SUPCRT convention.
hpha02ver.dat - The data in this file, provided courtesy of Oliver Jagoutz, a brilliant, tenureable, assistant professor at MIT, are identical to hp02ver.dat except that it includes: 1) a piecewise linear fit to the properties of quartz (Ohno et al., Phys Chem Min 33:1-9, 2006) that is intended to represent the anomalous behavior of quartz in the vicinity of the alpha-beta transition (the piecewise fit is manifest by four different quartz phases: q, q_anom, bq_anom, bq); and 2) linear fits to the adiabatic bulk and shear moduli data base of Hacker & Abers (2004). The virtues of this data base are that: 1) the elastic properties of quartz are poorly represented by Holland & Powell's Landau model as implemented in the original hp02ver.dat data file; and the Hacker & Abers (2004) compilation of elastic moduli is less haphazard than that in hp02ver.dat. The compilation is recommended particularly for applications involving quartz-bearing rocks. To use the linear models for bulk modulus provided with this data base, as opposed to bulk moduli derived from Holland & Powell's equation of state, the keyword explicit_bulk_modulus must be set to true in perplex_option.dat. SUPCRT convention.
hp04ver.dat - The Holland & Powell (1998) data base as revised by the authors in 2004. The primary difference between the 2002 and 2004 revisions is that in the 2002 version the aluminosilicate triple point is at 3.8 kbar and 780 K consistent with Holdaway's (1971) estimate; in the 2004 revision the properties of the aluminosilicates have been adjusted to place the triple point at 4.4 kbar and 823 K to satisfy a petrological argument of Pattison (1992). It is doubtful that this change is otherwise significant. The hp04ver.dat file does not contain shear moduli and therefore cannot be used (without modification) for seismic velocity calculations. The hp02ver.dat file is appropriate for calculations at depths < 440 km. SUPCRT convention.
hp11ver.dat - The TC-DS610 version of the Holland & Powell (2011) data base, generated Nov 13, 2011. This data base is based on an equation of state that is intended for use over the entire pressure range from the Earth's surface to the core-mantle boundary. It includes some lower mantle phases, but, at present, provides a less complete model for the lower mantle than the Stixrude & Lithgow-Bertelloni (2011) data base. This data base is not consistent with most of the current solution models for crustal and upper mantle silicates provided by Holland & Powell and coworkers in THERMOCALC. hp11ver.dat does not include shear moduli and therefore cannot be used to compute seismic wave speeds (except sound speed). HSC convention.
hp622ver.dat - The TC-DS622 version of the Holland & Powell (2011) data base, generated. HSC convention.
koma06ver.dat - the Komabayashi & Omori (2006) data base for dense hydrous Mg-silicates. To use this data base set approx_alpha and Anderson-Gruneisen to False. SUPCRT convention.
sfo05ver.dat - Stixrude & Lithgow-Bertelloni (2005) augmented for mid- and lower-mantle phases as described by Khan et al. (2006). See the header section of this file for a list of solution models to be used with the data base.
stx08ver.dat - Xu et al. (2008). See the header section of this file for a list of solution models to be used with the data base.
stx11ver.dat - Stixrude & Lithgow-Bertelloni (2011). The most comprehensive data base for mantle phase relations and seismic wave-speed calculations. See the header section of this file for a list of solution models to be used with the data base. NOTE: This thermodynamic data base requires the solution model file named stx11_solution_model.dat.
sup92ver.dat - The SUPCRT database, Johnson et al (1992), database notation. SUPCRT convention.
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.
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 |
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 = num_{1} * name_{1} + num_{2} * name_{2} +....+ num_{j} * name_{j}
dnum_{1} dnum_{2} dnum_{3}
where name is the name of the entity to be created, num_{i} is a number or fraction (i.e., two numbers separated by a '/') indicating the stoichiometric proportion of an entity name_{i} that exists as a data entry in the second part of the data file. The Gibbs energy of the made entity is
G_{name} = num_{1}·G_{name1} + num_{2}·G_{name2} +....+ num_{j}·G_{name}_{j} + G_{dqf} [2]
and
G_{dqf}[J/mol] = dnum_{1} + T[K]·dnum_{2} + P[bar]·dnum_{3} [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.
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, C_{P} (e.g., Helgeson et al. 1978; Berman 1988)
2 - normal polynomials fpr α, C_{P}, K_{T}; Murnaghan for V (e.g., Holland & Powell 1998)
3 - normal polynomial for C_{p}; Ghiorso (2003) polynomial on α, K_{T}; Birch-Murnaghan on pressure
4 - normal polynomials for α, C_{P}, K_{T}; 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, C_{P}; exponential polynomial for V (e.g., Haas et al. 1986; Gottschalk 1997)8 - normal polynomial for C_{p}; Mie-Gruneisen Einstein oscillator for thermal pressure; modified Tait for V (Holland & Powell, 2010)
9 - normal polynomials for C_{p}, α, K_{T}; Tait for V (Holland & Powell, 2010)
10 - ideal gas, normal polynomial for C_{p}
11 - Mie-Gruneisen Liquid (e.g., Stixrude et al. 2009)
12 - Brosh/CALPHAD format
13 - normal polynomial for C_{p}; Komabayashi & Omori (2006) polynomial on α, K_{T}, 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
name_{1}(num_{1})name_{2}(num_{2})name_{3}(num_{3}) ....
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 |
where name_{i} is the case-sensitive name of a component and num_{i} 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.
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 (P_{r}) and temperature (T_{r}) of the data base.
The values of c1-c7 correspond to the coefficients of the polynomial
C_{P}(T,P_{r}) = c1 + c2·T + c3/T^{2} + c4·T^{2} + c5/T^{1/2} + c6/T + c7/T^{3} + c8·T^{3} [4]
that describes the isobaric heat capacity (C_{P}) at the reference pressure.
The values of b1-b8 characterize the pressure-temperature dependence of the volume (V).
Volume (V, J/bar) at pressure (P) and temperature (T) is
V(T,P) = V0 + b2·(T-T_{r}) + b4·(P-P_{r}) + b6·(P-P_{r})^{2} + b7·(T-T_{r})^{2} [5]
Volume at pressure and temperature is
V(T,P) = V0·exp[b3·(T-T_{r}) + b8·(P-P_{r})] [6]
The isobaric expansivity (α) at the reference pressure is
α(T,P_{r}) = b1 + b2·T + b3/T + b4/T^{2} + b5/T^{1/2} [7]
that describes the isobaric expansivity (α) at the reference pressure. Volume at the reference pressure is
V(T,P_{r}) = V0·[1 + integral(α(T,P_{r}),T=T_{r}..T)] [8]
if the perplex_option.dat keyword approx_alpha is T (the default), or
V(T,P_{r}) = V0·exp[integral(α(T,P_{r}),T=T_{r}..T)] [9]
if approx_alpha is F.
The isothermal bulk modulus is
K(T,P_{r}) = b6 + b7·(T-T_{r}) [10]
if the perplex_option.dat keyword Anderson-Gruneisen is F, or
K(T,P_{r}) = b6·exp[-K'·integral(α(T,P_{r}),T=T_{r}..T)] [11]
if Anderson-Gruneisen is T.
Volume at the reference pressure is computed as
V(T,P_{r}) = V0 + b1·(T-1673) [12]
and the isothermal bulk modulus is
K(T,P_{r}) = V(T,P_{r})/[b2 + b3·(T-1673)] [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,P_{r}) = b1 + b2·T + b3/T + b4/T^{2} [34]
The bulk modulus at the reference pressure is
K(T,P_{r}) = 1/(b5 + b6·T + b7·T^{2} + b8·T^{3}) [35]
and its pressure derivative is
K' = b9 + b10·(T-T_{r})· ln(T/T_{r}). [36]
V(T,P_{r}) is computed as in equation [9] and V(T,P) is computed as in equation [14] below.
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,P_{r})·[1 - K'·P/{K'·P + K(T,P_{r})}]^{1/}^{K'} [14]
is used. Otherwise if b8 < -3, b8 = -K' and the Birch-Murnaghan equation
P = 3·K(T,P_{r})·f·[1 + 2·f]^{5/2}·[1 + {3·(K'-4)/4}·f], [15]
where
f =[{V(T,P_{r})/V(T,P)}^{2/3} - 1]/2, [16]
is used and solved numerically for V(T,P).
Documentation pending publication of Holland & Powell (2010, JMG)
EoS 8
EoS 9
The adiabatic shear modulus (bar) is
μ_{S}(T,P) = m0 + m1·(P-P_{r}) + m2·(T-T_{r}) [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
μ_{S}(T,P) = ν·K_{S}(T,P) [18]
where the Poisson ratio ν is specified by the second value of the poisson_ratio keyword and the adiabatic bulk modulus K_{S}(T,P) is computed by differentiation of the Gibbs energy (Equation 4 of Connolly & Kerrick 2002).
By default, Perple_X computes the adiabatic bulk modulus K_{S}(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 K_{S}(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 K_{S}(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
K_{S}(T,P) = k0 + k1·(P-P_{r}) + k2·(T-T_{r}). [18.1]
This explicit expression for the K_{S}(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 K_{S}(T,P) is computed by differentiation of the Gibbs energy.
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 = Θ_{0}·exp{-γ_{0}[{V/V_{0}}^{q0 }- 1]/q_{0}} [20]
and in the Stixrude & Lithgow-Bertelloni (2005b) formulation
Θ = Θ_{0}·a_{2}·[1 + (a_{1} + a_{2}·f/2)·f]^{1/2} [21]
with
a_{1} = 6·γ_{0} [22]
a_{2} = a_{1}·[a_{1} - 2 - 3q_{0}] [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]:
The only difference from the above list in the parameter assignments for the Stixrude & Lithgow-Bertelloni (2005b, EoS=6) formulation is
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:
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:
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).
Perple_X has a number of specialized internal EoS, a resume of these follows
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
C_{P},_{dis} = d1 + d2·T + d3/T^{2} + d4·T^{2} + d5/T^{1/2} + d6/T + d7/T^{2} [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.
The data for a Lambda (and mock-Lambda) transition of a thermodynamic entity is entered on a single line with the general format
transition = num_{1} type = num_{2} t1 = num_{3} ... t11 = num_{13}
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:
In the UBC lambda model, the lambda component of the Gibbs energy is evaluated in terms of the variable T' = T + T_{d}, where T_{d} = t4·(P-P_{r}). The lambda component of the heat capacity is
C_{Pλ} = T'·(t1 + t2·T') [25]
over the interval from t7 - T_{d }to the smaller of T' or t3 - T_{d}. t5 and t6 approximate the ∂V/∂P and ∂V/∂T derivatives for the entity, and t8 is an energetic correction.
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
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
The Landau model is based on three parameters: t1 - the temperature T_{0 }at which the phase becomes fully disordered at the reference pressure; t2 - the maximum entropy change of disorder, S_{d}; t3 - the maximum volume change of disorder, V_{d}. The Gibbs energy change of ordering due to a lambda transition is
ΔG = h' - T·s' + integral(v',P=P_{r}..P) + G_{Landau} [26]
_{where }
h' = S_{d}·T_{0}·(Q_{0}^{2} - Q_{0}^{6}/3) [27]
s' = S_{d}·Q_{0}^{2} [28]
v' = V_{d}·Q_{0}^{2}·[1 - K'·P/{K'·P + K(T,P_{r})}]^{1/}^{K'}·[1 + integral(α(T,P_{r}),T=T_{r}..T)] [29]
T_{c} = T_{0} + P·V_{d}/S_{d} [30]
Q_{0} = (1 - T_{r}/T_{0})^{1/4} [31]
if T < T_{c}, then Q = (1 - T/T_{0})^{1/4}, else Q = 0 [32]
G_{Landau} = S_{d}·[(T - T_{c})·Q^{2} + T_{c}·Q^{6}/3)] [33]
and the unspecified properties are those of the completely disordered state.
Documentation pending publication of Holland & Powell (2011, JMG)
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