Thermodynamic equilibrium at heterogeneous pressure

Abstract

Recent advances in metamorphic petrology point out the importance of grain-scale pressure variations in high-temperature metamorphic rocks. Pressure derived from chemical zonation using unconventional geobarometry based on equal chemical potentials fits mechanically feasible pressure variations. Here, a thermodynamic equilibrium method is presented that predicts chemical zoning as a result of pressure variations by Gibbs energy minimization. Equilibrium thermodynamic prediction of the chemical zoning in the case of pressure heterogeneity is done by constrained Gibbs minimization using linear programming techniques. In addition to constraining the system composition, a certain proportion of the system is constrained at a specified pressure. Input pressure variations need to be discretized, and each discrete pressure defines an additional constraint for the minimization. The Gibbs minimization method provides identical results to a geobarometry approach based on chemical potentials, thus validating the inferred pressure gradient. The thermodynamic consistency of the calculation is supported by the similar result obtained from two different approaches. In addition, the method can be used for multi-component, multi-phase systems of which several applications are given. A good fit to natural observations in multi-phase, multi-component systems demonstrates the possibility to explain phase assemblages and zoning by spatial pressure variations at equilibrium as an alternative to pressure variation in time due to disequilibrium.

Appendices

Appendix 1: Gibbs energy calculation

Pure phases

Partial molar Gibbs energy (g ⁰) for mineral and liquid end-members is generalized as a linear combination of the independent (idp) set of end-members in the internally consistent thermodynamic data set of Holland and Powell (1998), hereafter referred to as HP98 data set. Enthalpy of formation of ordered end-members (Holland and Powell 1996) and energies for fictive end-members following from application of Darken’s Quadratic Formalism (Powell 1987; Holland and Powell 1992; Will and Powell 1992) are generalized by a single DQF parameter:

$$g^{0} = \sum\limits_{idp = 1}^{nidp} {v_{idp} \cdot } \Delta G_{idp}^{0} + {\text{DQF}}$$

(9)

For the independent end-members in the HP98 data set, the stoichiometric coefficient and the DQF parameter are 1 and 0, respectively. Partial molar Gibbs energy of these end-members is calculated following standard formulation (Spear 1993), with the addition of two excess Gibbs energy terms for phases treated to a Landau model (Holland and Powell 1998):

$$\Delta G^{0} = \underbrace {{\Delta_{f} H_{\text{ref}} + \int\limits_{{T_{\text{ref}} }}^{T} {C_{p} \cdot {\text{d}}T} + \int\limits_{{P_{\text{ref}} }}^{P} {V \cdot {\text{d}}P} }}_{H} - T \cdot \underbrace {{\left( {S_{\text{ref}} + \int\limits_{{T_{\text{ref}} }}^{T} {\frac{{C_{p} }}{T} \cdot {\text{d}}T} } \right)}}_{S} + \Delta G_{\text{exc}} + \Delta G_{\text{landau}}$$

(10)

Heat capacity is given by the polynomial in the caption of Table 5 in Holland and Powell (1998):

$$C_{p} = a + b \cdot T + c \cdot T^{2} + d \cdot T^{ - 1/2}$$

(11)

Volume at reference pressure (1 bar) and elevated temperatures is given by Holland and Powell (1998):

$$V_{1,T} = V_{\text{ref}} \cdot \left[ {1 + a^{0} \cdot \left( {T - T_{\text{ref}} } \right) - 20 \cdot a^{0} \cdot \left( {\sqrt T - \sqrt {T_{\text{ref}} } } \right)} \right]$$

(12)

Volume at elevated pressure is modeled with the Murnaghan equation of state rearranged for volume:

$$V = \frac{{V_{1,T} }}{{\left( {\frac{4 \cdot P}{{k_{T} + 1}}} \right)^{1/4} }}$$

(13)

The bulk modulus is given by Holland and Powell (1998):

$$k_{T} = k_{\text{ref}} \cdot \left( {1 - 1.5 \cdot 10^{ - 4} \cdot \left( {T - T_{\text{ref}} } \right)} \right)$$

(14)

Landau model excess energy

Phases undergoing order–disorder or lambda heat capacity anomalies are treated with a Landau model (Holland and Powell 1990, 1998). The excess Gibbs energy term related to this model is calculated from:

$$\Delta G_{\text{exc}} = h_{\text{ref}}^{'} - T \cdot s_{\text{ref}}^{'} + \int\limits_{{P_{\text{ref}} }}^{P} {v_{T}^{'} \cdot {\text{d}}P}$$

(15)

With the enthalpy and entropy at reference conditions given in Eqs. (16) and (17):

$$h_{\text{ref}}^{'} = S_{\hbox{max} } T_{c}^{0} \cdot \left( {Q_{\text{ref}}^{2} - \frac{1}{3} \cdot Q_{\text{ref}}^{6} } \right)$$

(16)

$$s_{\text{ref}}^{'} = S_{\hbox{max} } \cdot Q_{\text{ref}}^{2}$$

(17)

The volume integral is evaluated again with the Murnaghan equation of state using the volume at reference pressure and elevated temperature from:

$$v_{T}^{'} = \frac{{V_{\hbox{max} } \cdot Q_{\text{ref}}^{2} }}{{V_{\text{ref}} }} \cdot V_{1,T}$$

(18)

The Landau excess energy is then obtained with:

$$\Delta G_{\text{Land}} = S_{\hbox{max} } \left( {\left( {T - T_{c} } \right) \cdot Q^{2} + \frac{1}{3} \cdot T_{c} \cdot Q^{6} } \right)$$

(19)

This term is added only at temperatures below the critical temperature T _c :

$$T_{c} = T_{c}^{0} + \frac{{V_{\hbox{max} } }}{{S_{\hbox{max} } }} \cdot \left( {P - P_{\text{ref}} } \right)$$

(20)

The order parameter Q (and Q _ref evaluated at reference conditions) is calculated as:

$$Q = \left( {1 - \frac{T}{{T_{c} }}} \right)^{1/4}$$

(21)

Parameters needed in Eqs. (10)–(21) are: Δ _f H _ref, S _ref, a, b, c, d, V _ref, a ⁰, k _ref, Tc ⁰, S _max and V _max. Most updated values of these parameters are found in the tc-ds55 file bundled with the most recent version of THERMOCALC (http://www.metamorph.geo.uni-mainz.de/thermocalc). Reference conditions are 298.15°K at 1 bar. See Table 3 for a complete list of symbols and parameters used in “Appendix 1”.

Table 3 Description of symbols and parameters used in the presented equations in the Appendix 1

Ordered and fictive end-members

Stoichiometric coefficients (v _idp ) and name of independent end-members in Eq. 9 are found in activity–composition (a–x) files bundled with THERMOCALC or individually downloadable from http://www.metamorph.geo.uni-mainz.de/thermocalc. The DQF parameter capturing both the DQF energy of fictive end-members and the enthalpy of reactions forming ordered end-members is given as:

$${\text{DQF}} = a_{\text{DQF}} + b_{\text{DQF}} \cdot T + c_{\text{DQF}} \cdot P$$

(22)

Parameters a _DQF, b _DQF and c _DQF are found in the lines below the stoichiometric coefficients in the same a–x files.

Solid solutions

The partial molar Gibbs energy of mixing in solid solutions and melts consist of a mechanical (mech), ideal (id), and a non-ideal (nid) part:

$$g = g_{\text{mech}} + g_{\text{id}} + g_{\text{nid}}$$

(23)

Mechanical mixing Gibbs energy consists of a linear combination of the total number (np) of end-member Gibbs energies in the solution, obtained from Eq. (9) above, multiplied by its proportion p.

$$g_{\text{mech}} = \sum\limits_{ip = 1}^{np} {g_{ip}^{0} } \cdot p_{ip}$$

(24)

Ideal mixing Gibbs energy, or configurational energy, is obtained from the sum of crystallographic site fractions following Stirling’s approximation (see also the Appendix in Tajčmanová et al. 2009).

$$g_{id} = T \cdot \left( {R \cdot \sum\limits_{iz = 1}^{nz} {m_{iz} \cdot z_{iz} \cdot \ln \left( {z_{iz} } \right)} - \sum\limits_{ip = 1}^{np} {s_{ip}^{0} } \cdot p_{ip} } \right)$$

(25)

Definition of certain solution models results in nonzero site fractions for some end-members leading to nonzero configurational entropy for the pure end-member. This is corrected for by the last sum in Eq. (25), because in principle pure phases do not contribute to ideal mixing energy. The configurational entropy for the pure end-members in the solution is calculated from site fractions of pure end-member (z ⁰):

$$s_{ip}^{0} = R \cdot \sum\limits_{iz = 1}^{nz} {m_{iz} \cdot z_{ip,iz}^{0} \cdot \ln \left( {z_{ip,iz}^{0} } \right)}$$

(26)

For example, the anorthite end-member in ternary feldspar is defined as having the tetrahedral site filled half with Al and the other half with Si, so that both site fractions (Si and Al on the tetrahedral site) are half. An equivalent approach for chemical potentials is described in Powell and Holland (1993).

Non-ideal mixing Gibbs energy is generalized to account for ternary interaction parameters and asymmetric Van Laar formulation (Holland and Powell 2003):

$$g_{nid} = \sum\limits_{iw = 1}^{nw} {W_{iw}^{*} \cdot \prod\limits_{ip = 1}^{ni} {\varphi_{{wi_{ip,iw} }} } }$$

(27)

The Margules parameters $W_{iw}^{*}$ are multiplied by the product of proportions φ corresponding to the iwth interaction parameter. The indices of φ for each iwth interaction parameter are stored in a matrix “wi.” This matrix has nw number of Margules parameters and ni number of end-member indices. The number of multiplied proportions (ni) is depending on the solution model. Usually the Margules parameters are binary interaction, but for feldspar they are ternary interaction parameters. This corrected Margules parameter $W_{iw}^{*}$ is obtained from Margules parameters W _iw fitted in experiments (e.g., found in literature) multiplied by proportions of interacting end-members and corrected by a size parameter (α):

$$W_{{^{iw} }}^{*} = W_{iw} \cdot \frac{{ni \cdot \left( {\sum\nolimits_{k = 1}^{np} {\alpha_{k} \cdot p_{k} } } \right)}}{{\sum\nolimits_{ip = 1}^{ni} {\alpha_{{wi_{ip,iw} }} } }}$$

(28)

The asymmetric proportion φ of the ith end-member is found from:

$$\varphi_{i} = \frac{{p_{i} \cdot \alpha_{i} }}{{\sum\nolimits_{j = 1}^{np} {p_{j} \cdot \alpha_{j} } }}$$

(29)

Both the Margules and α parameters are in principle pressure and temperature dependent and parameterized as:

$$W = W_{0} + W_{T} \cdot T + W_{P} \cdot P$$

(30)

$$\alpha = \alpha_{0} + \alpha_{T} \cdot T + \alpha_{P} \cdot P$$

(31)

Values for α ₀ , α _Τ , α _P and W ₀ , W _T , W _P are found in literature describing solution models, from a–x files bundled with THERMOCALC or from the solution_model.dat file packaged with Perple_X.

Site speciation

Finding the site fractions as function of mineral compositions is done by setting up a linear system of equations. The first set of equations is found by the definition that site fractions on each site sum up to 1. This gives ns (number of sites) equations. For each isth site, the equation is:

$$\sum\limits_{i = 1}^{nz} {z_{is,iz} \cdot v_{is,iz} } = 1$$

(32)

The second set of equations is given by the definition of compositional variables:

$$\sum\limits_{iz = 1}^{nz} {z_{iz} \cdot v_{ic,iz} } = C_{ic}$$

(33)

In case of order–disorder in a mineral, for each ordered end-member, an extra equation is required that defines the order parameter.

$$\sum\limits_{iz = 1}^{nz} {z_{iz} \cdot v_{iQ,iz} } = Q_{iQ}$$

(34)

Then, any extra assumptions form additional equations (e.g., equal distribution of an element over different sites):

$$\sum\limits_{iz = 1}^{nz} {z_{iz} \cdot v_{iex,iz} } = 0$$

(35)

If the system of equations is not closed, a charge balance equation can be added to ensure electro-neutrality:

$$\sum\limits_{iz = 1}^{nz} {z_{iz} \cdot m_{iz} \cdot \varepsilon_{iz} } = 2 \cdot nO$$

(36)

After the site fractions have been found as function of compositional variables, the proportions of end-members as function of site fractions can be solved from the obtained matrix of site fractions and proportions of end-members.

The first equation is always the constraint that proportions sum up to 1:

$$\sum\limits_{ip = 1}^{np} {p_{ip} } = 1$$

(37)

For the remaining equations, the independent set of equations of site fraction as function of proportion is chosen:

$$z_{ind} = \sum\limits_{ip = 1}^{np} {p_{ip} \cdot \nu_{ind,ip}^{p} }$$

(38)

Appendix 2: Site speciation and proportion calculation example

Clinoamphibole

For clinoamphibole from Diener et al. (2007) (closely resembling orthoamphibole from the same authors), identical to cAmph(DP) or cAmph(DP2) in Perple_X, the crystallography can be tabulated as:

	Site 1		Site 2		Site 3				Site 4				Site 5
Crystallography	A		M13		M2				M4				T1
Multiplicity	1		3		2				2				4 (1)
Site fraction	z 1	z 2	z 3	z 4	z 5	z 6	z 7	z 8	z 9	z 10	z 11	z 12	z 13	z 14
Charge	0	1	2	2	2	2	3	3	2	2	2	1	4	3
Element	V	Na	Mg	Fe	Mg	Fe	Al	Fe3	Ca	Mg	Fe	Na	Si	Al

The brackets indicate a multiplicity which is employed by Diener et al. (2007) to calculate ideal mixing energy rather than using the actual site multiplicity (see also comments in solution_model.dat file from the current version of Perple_X software package). For charge balance in equations below, the correct multiplicity (4) is used.

From this the set of equations, to find site fraction can be written:

(39)

The last two equations to close the system are unknown site fractions that need to be varied independently. They function like an order–disorder parameter such as Q’s used in THERMOCALC formulations (Eq. 34).

Solving this for site fraction z gives:

$$\begin{aligned} & z_{1} = \frac{9}{2} - \frac{3}{4}{\text{Na}} - \frac{1}{2}{\text{Mg}} - \frac{1}{2}{\text{Ca}} - \frac{1}{2}{\text{Fe}} - \\\frac{1}{4}{\text{Fe}}^{3 + } - \frac{1}{4}{\text{Al}} \\ & z_{2} = \frac{3}{4}{\text{Na}} - \frac{7}{2} + \frac{1}{2}{\text{Mg}} + \frac{1}{2}{\text{Ca}} + \frac{1}{2}{\text{Fe}} + \frac{1}{4}{\text{Fe}}^{3 + } + \frac{1}{4}\text{Al} \\ & z_{3} = 1 - z_{\text{Fe}}^{{{\text{M}}_{13} }} \\ & z_{4} = z_{\text{Fe}}^{{{\text{M}}_{13} }} \\ & z_{5} = - \frac{3}{4} + \frac{1}{4}{\text{Mg}} + \frac{1}{4}{\text{Ca}} + \frac{1}{4}{\text{Fe}} - \frac{1}{8}{\text{Fe}}^{3 + } - \frac{1}{8}{\text{Al}} + \frac{1}{8}{\text{Na}} - z_{\text{Fe}}^{{{\text{M}}_{2} }} \\ & z_{6} = z_{\text{Fe}}^{{\text{M}_{2} }} \\ & z_{7} = \frac{7}{4} - \frac{1}{4}{\text{Mg}} - \frac{1}{4}{\text{Ca}} - \frac{1}{4}{\text{Fe}} - \frac{3}{8}{\text{Fe}}^{3 + } + \frac{1}{8}{\text{Al}} - \frac{1}{8}{\text{Na}} \\ & z_{8} = \frac{1}{2}{\text{Fe}}^{3 + } \\ & z_{9} = \frac{1}{2}{\text{Ca}} \\ & z_{10} = - \frac{3}{4} + \frac{3}{2}z_{\text{Fe}}^{{{\text{M}}_{13} }} + z_{\text{Fe}}^{{{\text{M}}_{2} }} - \frac{1}{4}{\text{Fe}} - \frac{1}{8}{\text{Na}} + \frac{1}{4}{\text{Mg}} - \frac{1}{4}{\text{Ca}} + \frac{1}{8}{\text{Fe}}^{3 + } + \frac{1}{8}{\text{Al}} \\ & z_{11} = - \frac{3}{2}z_{\text{Fe}}^{{{\text{M}}_{13} }} - z_{\begin{subarray}{l} {\text{Fe}} \\ \end{subarray} }^{{\text{M}_{2} }} + \frac{1}{2}\text{Fe} \\ & z_{12} = \frac{7}{4} + \frac{1}{8}{\text{Na}} - \frac{1}{4}{\text{Mg - }}\frac{1}{4}{\text{Ca}} - \frac{1}{4}{\text{Fe}} - \frac{1}{8}{\text{Fe}}^{3 + } - \frac{1}{8}{\text{Al}} \\ & z_{13} = \frac{15}{8} - \frac{1}{8}{\text{Mg}} - \frac{1}{8}{\text{Ca}} - \frac{1}{8}{\text{Fe}} - \frac{3}{16}{\text{Fe}}^{3 + } - \frac{3}{16}{\text{Al}} - \frac{1}{16}{\text{Na}} \\ & z_{14} = - \frac{7}{8} + \frac{1}{8}{\text{Mg}} + \frac{1}{8}{\text{Ca}} + \frac{1}{8}{\text{Fe}} + \frac{3}{16}{\text{Fe}}^{3 + } + \frac{3}{16}{\text{Al}} + \frac{1}{16}{\text{Na}} \\ \end{aligned}$$

(40)

Substituting the compositional variables (Al, Fe, Mg, Ca, Na and Fe³⁺) for each end-member (found in the tc-ds55 database file) and the site fractions for the ordered end-members (a and b) gives the site fractions in the table below:

		z 1	z 2	z 3	z 4	z 5	z 6	z 7	z 8	z 9	z 10	z 11	z 12	z 13	z 14
tr	p 1	1	0	1	0	1	0	0	0	1	0	0	0	1	0
ts	p 2	1	0	1	0	0	0	1	0	1	0	0	0	½	½
parg	p 3	0	1	1	0	½	0	½	0	1	0	0	0	½	½
gl	p 4	1	0	1	0	0	0	1	0	0	0	0	1	1	0
cumm	p 5	1	0	1	0	1	0	0	0	0	1	0	0	1	0
grun	p 6	1	0	0	1	0	1	0	0	0	0	1	0	1	0
a	p 7	1	0	1	0	0	1	0	0	0	0	1	0	1	0
b	p 8	1	0	0	1	1	0	0	0	0	0	1	0	1	0
mrb	p 9	1	0	1	0	0	0	0	1	0	0	0	1	1	0

Choosing the independent equations from the columns in the table above along with the requirement that the end-members sum up to one gives:

$$\begin{aligned} & 1 = p_{1} + p_{2} + p_{3} + p_{4} + p_{5} + p_{6} + p_{7} + p_{8} + p_{9} \\ & z_{1} = p_{1} + p_{2} + p_{4} + p_{5} + p_{6} + p_{7} + p_{8} + p_{9} \\ & z_{3} = p_{1} + p_{2} + p_{3} + p_{4} + p_{5} + p_{7} + p_{9} \\ & z_{5} = p_{1} + \frac{1}{2}p_{3} + p_{5} + p_{8} \\ & z_{6} = p_{6} + p_{7} \\ & z_{7} = p_{2} + \frac{1}{2}p_{3} + p_{4} \\ & z_{9} = p_{1} + p_{2} + p_{3} \\ & z_{10} = p_{5} \\ & z_{11} = p_{6} + p_{7} + p_{8} \\ \end{aligned}$$

(41)

Solving for p gives:

$$\begin{aligned} & p_{1} = \frac{1}{2}z_{1} + z_{5} + z_{6} - z_{10} - z_{11} - \frac{1}{2} \\ & p_{2} = \frac{1}{2}z_{1} - z_{5} - z_{6} + z_{9} + z_{10} + z_{11} - \frac{1}{2} \\ & p_{3} = 1 - z_{1} \\ & p_{4} = z_{5} + z_{6} + z_{7} - z_{9} - z_{10} - z_{11} \\ & p_{5} = z_{10} \\ & p_{6} = 1 - z_{3} + z_{6} - z_{11} \\ & p_{7} = z_{3} + z_{11} - 1 \\ & p_{8} = - z_{6} + z_{11} \\ & p_{9} = 1 - z_{5} - z_{6} - z_{7} \\ \end{aligned}$$

(42)

Appendix 3: Pressure constraints example

As an example, here the equations for the pressure constraints in a system with three different pressures, three phases and three compositions are spelled out.

The pressure constraints:

$$\begin{aligned} & \alpha_{1}^{1} + \alpha_{2}^{1} + \alpha_{3}^{1} = \pi_{\text{sys}}^{1} \\ & \alpha_{1}^{2} + \alpha_{2}^{2} + \alpha_{3}^{2} = \pi_{\text{sys}}^{2} \\ & \alpha_{1}^{3} + \alpha_{2}^{3} + \alpha_{3}^{3} = \pi_{\text{sys}}^{3} \\ \end{aligned}$$

(43)

The Gibbs energy function is to be minimized:

$$g_{\text{sys}} = \alpha_{1}^{1} \left. {g_{1} } \right|_{{P_{1} }} + \alpha_{2}^{1} \left. {g_{2} } \right|_{{P_{1} }} + \alpha_{3}^{1} \left. {g_{3} } \right|_{{P_{1} }} + \alpha_{1}^{2} \left. {g_{1} } \right|_{{P_{2} }} + \alpha_{2}^{2} \left. {g_{2} } \right|_{{P_{2} }} + \alpha_{3}^{2} \left. {g_{3} } \right|_{{P_{2} }} + \alpha_{1}^{3} \left. {g_{1} } \right|_{{P_{3} }} + \alpha_{2}^{3} \left. {g_{2} } \right|_{{P_{3} }} + \alpha_{3}^{3} \left. {g_{3} } \right|_{{P_{3} }}$$

(44)

The system composition constraints look similar to Eq. 44, for example, in case of having a component x ^CaO:

$$x_{\text{sys}}^{\text{CaO}} = \alpha_{1}^{1} x_{1}^{\text{CaO}} + \alpha_{2}^{1} x_{2}^{\text{CaO}} + \alpha_{3}^{1} x_{3}^{\text{CaO}} + \alpha_{1}^{2} x_{1}^{\text{CaO}} + \alpha_{2}^{2} x_{2}^{\text{CaO}} + \alpha_{3}^{2} x_{3}^{\text{CaO}} + \alpha_{1}^{3} x_{1}^{\text{CaO}} + \alpha_{2}^{3} x_{2}^{\text{CaO}} + \alpha_{3}^{3} x_{3}^{\text{CaO}}$$

(45)

For MATLAB, the constraint equations are written in matrix form resulting in:

$$\left[ {\begin{array}{*{20}c} {x_{1}^{{{\text{SiO}}_{2} }} } &\quad {x_{2}^{{{\text{SiO}}_{2} }} } & \quad {x_{3}^{{{\text{SiO}}_{2} }}}& \quad {x_{1}^{{{\text{SiO}}_{2} }} } &\quad {x_{2}^{{{\text{SiO}}_{2} }} } &\quad {x_{3}^{{{\text{SiO}}_{2} }} } &\quad {x_{1}^{{{\text{SiO}}_{2} }} } &\quad {x_{2}^{{{\text{SiO}}_{2} }} } &\quad {x_{3}^{{{\text{SiO}}_{2} }} } \\ {x_{1}^{\text{CaO}} } & \quad{x_{2}^{\text{CaO}} } & \quad {x_{3}^{\text{CaO}} } & \quad{x_{1}^{\text{CaO}} } &\quad {x_{2}^{\text{CaO}} } &\quad {x_{3}^{\text{CaO}} } &\quad {x_{1}^{\text{CaO}} } &\quad {x_{2}^{\text{CaO}} } &\quad {x_{3}^{\text{CaO}} } \\ {x_{1}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{2}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{3}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{1}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{2}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{3}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{1}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad {x_{2}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } & \quad{x_{3}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } \\ 1 &\quad 1 &\quad 1 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 & \quad0 & \quad1 &\quad 1 &\quad 1 &\quad 0 & \quad0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 & \quad0 & \quad0 &\quad 1 &\quad 1 &\quad 1 \\ 1 &\quad 1 & \quad1 & \quad1 &\quad 1 &\quad 1 &\quad 1 &\quad 1 &\quad 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\alpha_{1}^{1} } \\ {\alpha_{2}^{1} } \\ {\alpha_{3}^{1} } \\ {\alpha_{1}^{2} } \\ {\alpha_{2}^{2} } \\ {\alpha_{3}^{2} } \\ {\alpha_{1}^{3} } \\ {\alpha_{2}^{3} } \\ {\alpha_{3}^{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{\text{sys}}^{{{\text{SiO}}_{2} }} } \\ {x_{\text{sys}}^{\text{CaO}} } \\ {x_{\text{sys}}^{{{\text{Al}}_{2} {\text{O}}_{3} }} } \\ {\pi_{\text{sys}}^{1} } \\ {\pi_{\text{sys}}^{2} } \\ {\pi_{\text{sys}}^{3} } \\ 1 \\ \end{array} } \right]$$

(46)

The optimization algorithm (function linprog in MATLAB) then searches for the alpha’s between 0 and 1 that gives the minimum of Eq. 44 satisfying the equality matrix in Eq. 46. See Appendix 4 for code examples.

Appendix 4: Code examples

Code for standard P–T diagram calculation:

An example code to do minimization in presence of a pressure gradient: