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Chemical mass transfer in magmatic processes

I. Thermodynamic relations and numerical algorithms

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Abstract

Thermodynamic and mathematical relations are presented to facilitate the description of an algorithm for the calculation of chemical mass transfer in magmatic systems. This algorithm extends the silicate liquid solution model of Ghiorso et al. (1983) to allow for the quantitative modelling of natural magmatic processes such as crystal fractionation, equilibrium crystallization, magma mixing and solid-phase assimilation. The algorithm incorporates a new method for determining the saturation surface of a non-ideal multicomponent solid-solution crystallizing from a melt. It utilizes a mathematical programming (optimization) approach to determine the stable heterogeneous (solids+liquid) equilibrium phase assemblage at a particular temperature and pressure in magmatic systems both closed and open to oxygen. Closed system equilibria are computed by direct minimization of the Gibbs free energy of the system. Open system equilibria are determined by minimization of the Korzhinskii potential (Thompson 1970), where oxygen is treated as a perfectly mobile component. Magmatic systems undergoing chemical mass transfer processes are modelled in a series of discrete steps in temperature, pressure or bulk composition, with each step characterized by heterogeneous solid-liquid equilibrium. A numerical implementation of the algorithm has been developed (in the form of a FORTRAN 77 computer program) and calculations demonstrating its utility are provided in an accompanying paper (Ghiorso and Carmichael 1985).

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Abbreviations

\(a_{\varphi _i } \) :

the activity of the i th solid component in the solid phase ϕ

c :

number of components in (ϕ th solid phase (i.e., equivalent to m ϕ)

\(f_{{\text{O}}_{\text{2}} } \) :

the fugacity of oxygen in the system

G :

the Gibbs free energy of the system

f :

intermediate variable used in step 2 of algorithm 2

G L :

the Gibbs free energy of the liquid

\(G_{M_1 } ,G_{M_2 } ,...,G_{M_p } \) :

the Gibbs free energy of the 1st, 2nd, ..., p th solid

\(G_{O_2 } \) :

the Gibbs free energy of oxygen in the system

\(g_{\varphi _i } \) :

the partial derivative of \(a_{\varphi _i } \) with respect to \(X_{\varphi _i } \) evaluated at ϕ

h :

intermediate variable used in step 2 of algorithm 2

L :

the Korzhinskii potential of a system open to oxygen

l :

the number of liquid components

m 1, m 2, ..., m p :

the number of components in the 1st, 2nd, ..., p th solid

n :

the number of liquid and solid components in the system (i.e., l+m 1+m 2+...+m p)

\(n_{O_2 } \) :

the excess oxygen content of the system (see Eq. 36)

P :

pressure

p :

the number of solid phases in the system

R :

the universal gas constant

r :

the ratio of ferric iron/total iron in the liquid

T :

the absolute temperature

X ϕ :

the mole fraction of the i th solid component in the solid phase ϕ

SSQ:

intermediate variable used in step 7 of algorithm 2

y :

intermediate variable used in step 2 of algorithm 2

α :

steplength parameter of order unity defined by Eqs. (28) and (29)

\(\gamma _{\varphi _i } \) :

the activity coefficient of the i th solid component in the phase ϕ

ΔG i :

the free energy change for the i th solid component melting to form a liquid at a particular T and P

k :

the condition number of the projected Hessian of G (Htilde)

\(u_{{\text{O}}_{\text{2}} } \) :

the chemical potential of oxygen in the magma

\(u_{_{{\text{O}}_{\text{2}} } }^0 \) :

the standard state chemical potential of oxygen in the system

\(u_{\varphi _i }^0 \) :

the standard state chemical potential of the i th solid component in the solid phase ϕ

∑ϕ :

the saturation index for the ϕ th solid phase

τ :

a number defined to be 10−t where t is the number of significant digits desired in the computation

ϕ i :

the i th end-member component in the solid phase ϕ

b :

vector which describes the bulk composition of the system in terms of liquid components

g :

vector of chemical potentials of each component in the system

\(\widetilde{\text{g}}\) :

the projected gradient of the system

g L :

vector of chemical potentials of each component in the liquid

\({\text{g}}_{M_1 } {\text{,g}}_{M_2 } ,....,{\text{g}}_{M_p } \) :

vector of chemical potentials of each component in the 1st, 2nd, ..., pth solid

\({\text{g}}_{{\text{O}}_2 } \) :

the gradient of the Gibbs free energy of oxygen with respect to n

n :

vector of moles of all liquid and solid components in the system (n and n′ are particular guesses for n)

n L :

vector of moles of each component in the liquid

\({\text{n}}_{M_1 } {\text{,n}}_{M_2 } ,....,{\text{n}}_{M_p } \) :

vector of moles of each component in the 1st, 2nd, ..., pth solid

n 1, n 2 :

compositional vectors defined by Eq. (21) and which denote the constrained and unconstrained parts of the vector n

X ϕ :

vector of mole fractions of the c solid components in the ϕ th phase, i.e., \([X_{\varphi _1 } {\text{,}}X_{\varphi _2 } ,...,X_{\varphi _c } ]^T \) (\(\hat X_\varphi \) is a particular guess for Xϕ

λ :

vector of Lagrange multipliers for the active equality constraints in the system (Eq. 34)

v i :

stoichiometric reaction coefficients for each liquid component in the reaction which describes the dissolution of the i th solid component

C :

block matrix which embodies the bulk composition constraint on the system (Eq. 9)

H :

second derivative matrix (Hessian) of the Gibbs free energy of the system

\(\widetilde{\text{H}}\) :

the projected Hessian of the system

H L :

second derivative matrix (Hessian) of the free energy of the liquid

\({\text{H}}_{M_1 } {\text{,H}}_{M_2 } {\text{,}}...{\text{,H}}_{M_p } \) :

second derivative matrix (Hessian) of the Gibbs free energy of the 1st, 2nd, ..., p th solid solution

\({\text{H}}_{{\text{O}}_2 } \) :

second derivative matrix of the Gibbs free energy of oxygen

I l :

identity matrix of order l

K :

an orthogonal matrix which right diagonalizes C Eq. (18). It may be partitioned K T=[K 1∶K 2]T

R :

upper triangular matrix formed by right diagonalizing C. It may be partitioned R=[R 11∶ O](see Eqs. 18, 19)

\({\text{T}}_{M_1 } {\text{,T}}_{M_2 } {\text{,}}...{\text{,T}}_{M_p } \) :

matrix which transforms a mole vector for the 1st, 2nd, ..., p th solid from solid into liquid compositional variables

O /skj /i :

matrix of zeros with i rows and j columns

\(\left| {\hat n} \right.\) :

evaluated at \(\widehat{\text{n}}\)

n T :

the transpose of n

∥n∥:

the euclidian (or L 2) norm of n, i.e. ∥n∥=(n T n)1/2

ln:

natural (base e) logarithm

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  1. Department of Geological Sciences, University of Washington, 98195, Seattle, Washington

    Mark S. Ghiorso

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Ghiorso, M.S. Chemical mass transfer in magmatic processes. Contr. Mineral. and Petrol. 90, 107–120 (1985). https://doi.org/10.1007/BF00378254

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