Chemical heterogeneities in the mantle: The equilibrium thermodynamic approach

    Chemical Heterogeneities in the Mantle: the Equilibrium Thermodynamic Approach M. Tirone, S. Buhre, H. Schm¨uck, K. Faak PII: DOI: Reference:

S0024-4937(15)00438-7 doi: 10.1016/j.lithos.2015.11.032 LITHOS 3767

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Please cite this article as: Tirone, M., Buhre, S., Schm¨ uck, H., Faak, K., Chemical Heterogeneities in the Mantle: the Equilibrium Thermodynamic Approach, LITHOS (2015), doi: 10.1016/j.lithos.2015.11.032

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Chemical Heterogeneities in the Mantle: the Equilibrium Thermodynamic Approach M. Tironea,∗, S. Buhreb , H. Schm¨ uckb , K. Faaka

f¨ ur Geologie, Mineralogie und Geophysik, Ruhr-Universit¨ at Bochum, D-44780 Bochum, Germany b Institute of Geosciences, Johannes Gutenberg University, D-55128, Mainz, Germany

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Abstract

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This study attempts to answer a simple and yet fundamental question in relation to our understanding of the chemical evolution of deep Earth and planetary interiors. Given two initially separate assemblages

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(lithologies) in chemical equilibrium can we predict the chemical and mineralogical composition of the two assemblages when they are put together to form a new equilibrated system? Perhaps a common perception is that given sufficient time, the two assemblages will homogenize chemically and mineralogically, however

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from a chemical thermodynamics point of view, this is not the case. Certain petrological differences in terms of bulk composition, mineralogy and mineral abundance remain unless other processes, like melting

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or mechanical mixing come into play. While there is not a standard procedure to address this problem, in this study it is shown that by applying chemical thermodynamic principles and some reasonable assumptions, it

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is possible to determine the equilibrium composition of each of the two assemblages. Some examples that consider typical mantle rocks, peridotite, lherzolite, dunite and eclogite described by simplified chemical systems are used to illustrate the general approach. A preliminary application to evaluate the effect of melting an heterogeneous mantle in complete chemical equilibrium using a thermodynamic formulation coupled with a two-phase geodynamic model, show that major elements composition of the melt product generated by different peridotites is very similar. This may explain the relatively homogeneity of major elements of MORBs which could be the product of melting a relatively uniform mantle, as commonly accepted, or alternatively a peridotitic mantle with different compositions but in chemical equilibrium. Keywords: chemical thermodynamics, mantle chemical heterogeneities, Gibbs free energy minimization, mantle melting, numerical modeling, thermodynamics, geodynamics

∗ Corresponding

author Email addresses: [email protected] (M. Tirone), [email protected] (S. Buhre), [email protected] (H. Schm¨ uck), [email protected] (K. Faak)

Preprint submitted to Elsevier

December 15, 2015

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1. Introduction One of the fundamental challenges in solid Earth sciences is understanding the chemical evolution of

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the Earth’s deep interior. A great debate has been carried on for several decades over the nature of the

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chemical heterogeneities in the mantle, how these chemical variations evolve and over what period of time

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(e.g. Gurnis and Davies, 1986; Morgan and Morgan, 1999; Poirier, 2000; Schubert et al., 2001; Tackley and

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Xie, 2002; van Keken and Ballentine, 2002; Trampert et al., 2004; Dobson and Brodholt, 2005; Carlson,

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2006; Helffrich, 2006; Tolstikhin et al., 2006; Zhong, 2006). Geophysical evidence (e.g. van der Hilst et al.,

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1997) and numerical models (e.g. Christensen and Hofmann, 1994) followed by several studies afterwards

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provided indications that lithospheric material compositionally different from the surroundings may subduct

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in the lower asthenospheric mantle. Furthermore experimental studies (Kawazoe and Ohtani, 2006; Sakai et

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al., 2006; Ozawa et al., 2009) and arguments based on geophysical data (Garnero, 2004; Garnero and Mc-

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Namara, 2008) indicated that at the core-mantle boundary some chemical exchange may take place between

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the outer core and the lower mantle. Thermo-chemical erosion of cratonic roots and ancient lithosphere may

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also contribute to compositional variations of the asthenospheric mantle (Xu , 2001; Foley, 2008).

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The arguments in support or against the persistent presence of large chemical variations in the mantle

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through the Earth history are largely based on indirect evidence from geophysical and geochemical observa-

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tions. Understanding the chemical evolution of the mantle has also potential implications for the convective

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structure of the mantle that in the two most accepted hypotheses involves the whole mantle or a layered

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structure (Peltier and Jarvis, 1982; van der Hilst et al., 1997; van Keken and Ballentine, 1998; Butler, 2009).

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The need for a better insight to the problem comes also from the consideration that trace elements and iso-

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topic studies provides useful but not unequivocal information regarding chemical heterogeneities and mantle

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chemical evolution and these studies often ignore the complexity of the processes involved.

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Chemical homogenization in the mantle is most effectively the result of the combined action of chemical and

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mechanical mixing (Kellogg and Turcotte, 1987; Kellogg, 1992). Mechanical mixing has been investigated

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using numerical models, while chemical mixing has been often ignored based on the argument that chemical

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diffusion is extremely slow and ineffective over large distance (Hofmann and Hart, 1978; Kellogg and Tur-

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cotte, 1987; Kogiso et al., 2004).

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Equilibration of two lithologies has been addressed on a different scale by several studies in metamorphic

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petrology using the irreversible thermodynamic approach (Fisher, 1973; Joesten, 1977; Nishiyama, 1983;

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Markl et al., 1998), which can be applied, with some limitations, to describe chemically reactive systems

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involving diffusion controlled reactions between different lithologies. In general a critical point that need to

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be considered is the scale of the problem (Knapp, 1989; Zhu and Anderson, 2002). In metamorphic petrol-

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ogy detailed observations on a scale ranging from mm to m demand a description of the physico-chemical

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processes involved at the same detailed scale. Numerical transport models often in hydrogeology and always

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in geodynamics assume local thermodynamic equilibrium on a spatial and temporal scale defined by the

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conditions imposed in the numerical model. The true scale of equilibration depends on the kinetic of the

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processes involved, however practical considerations on the objective of the modelling and the observational

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evidence that are used to validate the model (if any) should be taken into account. For instance detailed

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mineralogical description of the processes at the interface of two rock assemblages is marginally important

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in a mantle dynamic model in which the spatial grid is on the order of km and the observations are based

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on seismological data or global petrological data. However if the temporal scale of the model span several

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hundreds of Myrs or more, then the extent of equilibration between two lithologies may have an influence on

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the outcome of the simulation at the spatial scale of the model and should not be ignored. Commonly this

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type of chemical interactions are not considered in geodynamic models, in other words if the model includes

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two lithologies, these lithologies are never modified, regardless of the timescale. In fact it is commonly

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assumed either complete disequilibrium (fixed compositions, e.g. subducting slab, lithospheric mantle, crust

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etc.) or complete equilibrium (defined by one composition for the whole mantle) where only melt can alter

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a predefined composition. These end-member scenarios are essentially correct at time zero or infinite time

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from the creation of the heterogeneity (the latter only under certain conditions briefly discussed in the

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Conclusions). Mass transfer induced by melting sometimes is approximately accounted for using particle

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tracers (Christensen and Hofmann, 1994; Walzer and Hendel, 1999; Huang and Davies, 2007; Brandenburg

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et al., 2008; Balmer et al., 2009, 2011). A better representation, although restricted to mass and chemical

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exchange between solid and melt, is offered by two-phase flow models under the assumption of local equi-

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librium between the melt and the residual solid.

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The condition of thermal and mechanical disequilibrium has been discussed from a thermodynamic stand-

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point by Bina and Kumazawa (1993). It should be also possible to apply the thermodynamic approach to

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study chemical heterogeneities in the Earth’s interior and eventually extends it to the evolution of dynamic

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systems by considering a sequence of local states in thermodynamic equilibrium. The same assumption

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applies to the fundamental description of transport of physico-chemical properties in a continuous medium

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(Bird et al., 2002).

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This study begins to tackle the problem of understanding the evolution of chemical heterogeneities in the

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solid mantle from a different perspective, that is by developing a procedure to describe chemical equilibration

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between two lithologies. Here this procedure is discussed in some details with the help of some examples

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in which the composition of two mantle assemblages in equilibrium has been determined. A preliminary

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application to mid-ocean ridges shows the results of melting an heterogeneous mantle in complete chemical

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equilibrium using a thermodynamic formulation combined with a two-phase dynamic model.

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A step forward towards addressing the timescale of chemical evolution of the mantle using geodynamic and

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petrological numerical studies will require to combine the procedure discussed in this study with a proper

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formulation that models experimental data on the kinetics of the chemical processes driving the systems

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towards equilibrium.

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2. Outline of the problem and study cases

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The problem addressed in this study is illustrated in Fig. 1. Two separate assemblages with different

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bulk composition defined by the oxides abundance are independently in chemical equilibrium at the same

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given pressure and temperature (P,T) conditions. Once the two assemblages are put together at the same

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conditions, the new system is defined by the bulk composition of the two assemblages in some predefined

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proportion (grey box in Fig. 1). For simplicity in the following sections, it is only considered a combined

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system in which the proportion of the two assemblages is 1:1, i.e. the new whole system is made of the

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sum of the oxides molar abundance of the two assemblages. Assuming that the new system is in complete

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chemical equilibrium, the objective is to find the composition and mineralogy of the two sub-systems or in

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other words find the compositional and mineralogical changes that took place in the two sub-assemblages

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after they have been brought together. A standard procedure to solve this problem is not available. However

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in the next sections few cases of different complexity are used to show how an approximate solution can

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be found and to illustrate the procedure to overcome certain difficulties inherent to each case. One of the

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thermodynamic tools applied to this problem is the Gibbs free energy minimization which determines the

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equilibrium assemblage in terms of mineral composition and abundance given a certain bulk composition

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and P,T conditions (van Zeggeren and Storey, 1970; Eriksson, 1971, 1973, 1975; Smith and Missen, 1991).

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The essential prerequisite is the knowledge or definition of the thermodynamic properties of the considered

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phase components in a particular compositional system. The thermodynamic database applied here, which

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is reasonably acceptable to describe certain ultramafic rocks in the compositional system N a2 O − CaO −

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M gO − F eO − Al2 O3 − SiO2 (or related sub-systems), can be found in Tirone et al. (2012). Name of

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the minerals, abbreviations and formulas are listed in Table 1. The exclusion of minor components from

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the model greatly simplify the thermodynamic treatment. The effect of these components is to stabilize

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accessory minerals that most likely have a relatively small influence on the major element composition of

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the most abundant minerals and in general limited effect on the average physical properties of the rock

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assemblage. However the incorporation of minor components in the bulk composition could potentially shift

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the P,T stability conditions of the predominant mineral phases (Ganguly, 2008).

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The pressure and temperature at which equilibration is assumed for all cases discussed in this study is 30

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kbar and 1100o C. The above conditions are not a limitation, in fact the method discussed here can be easily

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applied to other chemical systems at much higher pressure using the appropriate thermodynamic database.

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No fluid phase is involved or is considered in the present model. The main reason is the fluid mobility and

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the transient state associated to the presence of a fluid. If equilibrium between two lithologies and a fluid

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is imposed, then it must be assumed that the fluid is available for the entire duration of the equilibration

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process and fluid transport in and out of the whole system should be taken into consideration. Because

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of the difficulties of the modeling approach, in particular for large scale problems, and the uncertainty on

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determining the geological setting in which the assumption of long term fluid circulation is applicable, in

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the following treatment all thermodynamic systems are assumed under dry conditions.

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2.1. Case 1: peridotite-eclogite, system CaO−M gO−Al2 O3 −SiO2 , two assemblages with partially different

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mineralogy

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The system CaO − M gO − Al2 O3 − SiO2 is considered in this section to describe the approximate

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composition of a peridotite and eclogite initially defined as (A0 ) and (B0 ). Bulk compositions in weight

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% (wt%) and moles are given in Table 2. The eclogite bulk composition was defined by the composition

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of a melt obtained from a thermodynamic computation in which, using a peridotite with bulk composition

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defined by A0 , the equilibrium product at 15 kbar and 1700o C consisted of depleted peridotite and melt. The

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bulk compositions for the peridotite and eclogite are used in the Gibbs free energy minimization calculation

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at given P,T conditions (30 kbar and 1100o C) to find the two independent equilibrium assemblages (Table

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2). A third thermodynamic computation is done on the combined bulk composition (C) (sum in moles

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of the two bulk compositions), the equilibrium assemblage is also reported in Table 2. This is the average

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assemblage of the whole system which does not represent the minerals distribution or the mineral abundance

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in the two sub-systems after they are put together. These are yet to be determined and how it is done is

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the main focus of this study. From the initial set of thermodynamic results two observations can be made.

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The moles of cpx and gt in the average assemblage is the weighted sum of the moles in the two separated

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B C assemblages (e.g. nA gt + ngt = ngt , where ngt = npy + ngrs ). Furthermore the components fraction in cpx

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and gt in the two initial assemblages (A0 , B0 ) and in the average assemblage (C) are the same (Table 2).

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Chemical equilibrium between two systems is generally defined by the equality of the chemical potentials of

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B C their components (cf. Lewis and Randall, 1961; Ganguly, 2008), in this case for example µA di = µdi = µdi , and

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similarly for the other components in cpx and gt. Since in the thermodynamic model used in this study the

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solid mixtures are treated as ideal solutions (a reasonable approximation when the thermodynamic database

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is tuned to describe a restricted compositional range), equilibrium can be also verified by the identity of

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the component fractions. It follows that the mineral components in cpx and gt in the initial assemblages

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(A0 ) and (B0 ) after they have been put together are already in thermodynamic equilibrium and these

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mineral components will not take part to any chemical exchange between the two assemblages. Non-ideal

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mixtures would make the equilibrium analysis more elaborate, for example at equilibrium there will be no

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correspondence between the identity of the chemical potentials and the identity of the molar components

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in the two assemblages, but the overall considerations that follow would remain the same. Up to this point

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the two observations indicate that no chemical transformations involving these mineral phases take place,

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which implies that after equilibration of (A0 ) and (B0 ), the composition and abundance of gt and cpx on

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the peridotite and eclogite sides remain unmodified. Further insight into the chemical exchanges between

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the two sub-systems is provided by the set of independent chemical reactions in the whole system that can

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be found using a standard stoichiometric procedure (Smith and Missen, 1991). For this case 3 reactions can

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be found: [1] py + di ↔ cats + 4 en, [2] 2 en + grs ↔ 2 di + cats, [3] qz + f o ↔ 2 en. The first two reactions

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do not add any information since gt and cpx are already in equilibrium. In principle, transformations of cpx

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and gt components in one sub-system could be counterbalanced by similar transformations with opposite

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sign on the other sub-system so that the net result is zero and mass balance of the whole system (C) is

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preserved. These transformations would require a certain energy change in the two sub-systems. From a

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thermodynamic standpoint it could be acceptable to decrease locally the entropy as long as a larger increase

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occurs elsewhere in the system (Kondepudi and Prigogine, 1998). However in this case it doesn’t seems

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a likely scenario considering that the whole system can reach the same equilibrium energy state without

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local transformations, that is leaving the amount of gt and cpx unchanged in the two sub-systems. The

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last reaction indicates that qz on the eclogite side and fo on the peridotite side will be consumed to create

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en. Since qz is not present in the whole assemblage (C), all qz will be consumed (1.257 moles), an equal

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amount of fo will be consumed on the peridotite side and two times this amount of en will be created.

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Whether en will form on the peridotite or eclogite side is still undetermined. The exact distribution can

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only be evaluated by kinetic experimental studies. However for a given distribution of en in (A) and (B), it is

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possible to compute the Gibbs energy of the two sub-systems GA and GB after they have been equilibrated

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together. In figure 2 a plot of the change of the Gibbs energy in the two sub-systems normalized to the

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A A0 Gibbs energy of (A0 ) and (B0 ) as a function of the change of the moles of en in (A) ∆nA en = nen − nen

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shows that only in a small compositional range between ∼1.770 and ∼1.775 the Gibbs energy on both sides

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B increases with respect to the initial values GA 0 and G0 . The assumption made here is that the distribution

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of en falls within this region in particular at the point where the increase of the Gibbs energy given by the

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following expression is the smallest (Fig. 2):  A 2  B 2 (G − GA (G − GB 0 ) 0) + GA GB 0 0

(1)

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Because the Gibbs energy in the two sub-systems increases, then the entropy in the two sub-systems must

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decrease (Ganguly, 2008), therefore the distribution of en under isothermal conditions can be related indif-

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ferently to the least Gibbs energy change or the least entropy change. The final composition of the peridotite

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and eclogite for the whole system in complete chemical equilibrium is summarized in Table 2. While the

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distribution of en determined with the above procedure is clearly an approximation, it is interesting to look

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at the results of kinetic experiments on growth of pure en in a qz+fo system (Milke et al., 2007). Milke et

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al. (2007) defined a coefficient ν to quantify the amount of en forming on the qz and fo side. According to

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the system of reactions considered in their study, for ν equal to 1, the only mobile component is SiO2 and it

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B reacts with fo. Translating this process in the peridotite-eclogite system, ∆nA en = 2.515 and ∆nen = 0. The

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other end-member possibility is given by ν = 0, in this case only M gO would be mobilized and en would

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form in equal amounts on each side. For the problem discussed in this section, this end-member process

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B corresponds to ∆nA en = 1.257 and ∆nen = 1.257. The experimental study suggests a value for ν of ∼ 0.4,

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B which is equivalent to ∆nA en = 1.760 and ∆nen = 0.755. These values are remarkably close to those obtained

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B by the approximate thermodynamic method developed in this study (∆nA en =1.773 and ∆nen =0.742, Table

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2). However such interesting result should be taken with some caution considering that several factors like

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oxygen fugacity, pressure, and water may have an effect on the value of ν and on the location where enstatite

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would form (Milke et al., 2007). Nevertheless it should be kept in mind that the approach proposed here

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does not aim to describe the detailed microstructural developments at the mineral scale but only to provide

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an approximate method that can be applied for the interpretation of chemical transformations on a large

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scale and for geodynamic applications.

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2.2. Case 2: peridotite-eclogite, system N a2 O −CaO −M gO −Al2 O3 −SiO2 , two assemblages with different

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mineralogy

In this section the chemical system includes also the sodium component. The two assemblages are once

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again a peridotite (A0 ) and an eclogite (B0 ). Like in the previous case, the eclogite composition is defined

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by the melt phase obtained from melting a peridotite but at slightly different conditions, P=15 kbar and

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T=1550o C. Bulk and mineralogical compositions at equilibrium of (A0 ) and (B0 ) are given in Table 3. It

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can be noticed in (B0 ) the coexistence of fo + qz instead of en. The combination is thermodynamically

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permissible, although rare. For example in the CMAS system with high aluminum content enstatite is not

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stable at high pressure (Presnall, 1999). Once the two bulk compositions (A0 ) and (B0 ) are put together

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in equal proportion, the new average assemblage (C) obtained from the Gibbs energy minimization reveals

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several differences when compared with the two initial assemblages. The original en in peridotite and

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qz in eclogite are completely consumed, sp is formed, and the component fractions in cpx and gt are all

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different in (A0 ), (B0 ) and (C) (Table 3). The stoichiometric analysis determines 4 independent reactions:

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[1] di + sp ↔ cats + f o, [2] di + py ↔ cats + 4 en, [3] gr + 2 en ↔ 2 di + cats and [4] qz + f o ↔ 2 en. To find

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the mineralogical composition and mineral abundance on the peridotite and eclogite side after equilibration,

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the reactions can be used to constrain the mineral changes on the two sides by relating these changes to

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the extent of the reaction ξ defined for each reaction (Prigogine and Defay, 1954; Kondepudi and Prigogine,

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1998). The extent of the reaction can be used to describe the changes of moles of a particular component j

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in a phase i, n(i,j)r − n0(i,j)r = νr ξr , where νr is the stoichiometric coefficient for the component ni,j in the

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reaction r. In this way, for instance the total molar change of di can be related to the extent of the reactions

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[1],[2] and [3]:

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B ∆nA di + ∆ndi − ξ1 − ξ2 + 2 ξ3 = 0

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and similar relations can be written for cats, en, fo, sp, py, gr and qz. In addition mass balance relations

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for each component can be expressed like for the di component:

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B C ∆nA di + ∆ndi = ∆ndi

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The condition of thermodynamic equilibrium requires that the chemical potential µ of all the components

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on the two sides of the whole system should be equal (Ganguly, 2008), for the di component: B C µA di = µdi = µdi

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(4)

As mentioned in the previous section, the thermodynamic database used in this study considers ideal mixing

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models to describe the solid solutions, hence equation 4 is equivalent to:

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(5)

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B C xA di = xdi = xdi

where xdi is the molar fraction of the component di. Expressions similar to equation 5 can be defined for all

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the components in cpx and gt. The final requirement is that the total Gibbs energy of (C) should be equal

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to the combination of the total Gibbs energy of (A) and (B):

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(6)

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GA + GB = GC

The above constraints (equations 2, 3, 5, 6) together with the least change thermodynamic assumption given

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by equation 1 are applied to determine the molar changes in the two sub-assemblages (A) and (B) and the

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extent of the reactions. Two additional limitations have been imposed to the possible solutions. The jd

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component on both sides is fixed since there are no other sodium bearing phases in the thermodynamic

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database at the given P,T conditions. Sp is allowed to form only on the eclogite side because reaction

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[1] indicates that formation of sp is associated to the formation of di and several computational attempts

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showed that positive change of di occurs only in the sub-assemblage (B). The numerical solution to this

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problem is not straightforward since it would involve a multi-objective optimization typically studied using

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evolutionary algorithms (e.g. Deb, 2001). Here a simpler approach has been adopted using a general purpose

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minimization program (MINUIT, James (1994)) to optimize a single cumulative function that includes all

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the constraint equations. Uniqueness and convergence of the solution has been validated by starting the

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minimization procedure with randomly chosen initial values for the fitting parameters. The outcome of this

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procedure is the composition of the two sub-assemblages (A) and (B) in equilibrium that satisfies all the

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above constraints (Table 3). The by-products of the minimization are the extent of the reactions [1]-[4]:

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ξ1 =1.354, ξ2 =5.206, ξ3 =1.148, ξ4 =-8.281. The values of ξ have no particular relevance for this study. It

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is evident from Table 3 that the two assemblages in chemical equilibrium (A) and (B) are substantially

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different in terms of mineralogical abundance and bulk composition, and we should expect differences in

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their physical properties as well.

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2.3. case 3: peridotite-lherzolite, system N a2 O − CaO − M gO − Al2 O3 − SiO2 , two assemblages with same mineralogy The equilibration of a fertile peridotite with a depleted peridotite is explored in this section. The bulk

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composition of peridotite (A0 ) is the same as the one in section 2.2. The bulk composition of the depleted

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peridotite (B0 ) is chosen in a way that in the equilibrium assemblage the same set of mineral phases and

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components are present in (A0 ) and (B0 ). The bulk compositions and the separate equilibrium assemblages

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are shown in Table 4. In the combined system (C) the possible reactions are: [1] di+py ↔ cats+4en and [2]

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en + grs ↔ cats + 2 di. Fo does not enter in any reaction, in fact in the average assemblage (C), it is simply

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the sum of fo in (A0 ) and (B0 ). As in the previous case, the jd component in cpx remains unmodified. The

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same general constraints discussed earlier and similar to equations 2, 3, 5, 6, are applied here to retrieve the

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composition and mineralogical abundance in (A) and (B) after equilibration. However in this problem the

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set of constraints are not sufficient to determine a unique solution for the mineralogy in the peridotite and

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eclogite sub-systems. Fig. 3 shows a range of compositions (change of moles of en in (A) ∆nA en and moles of

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gt in (A) and (B)) that are consistent with the imposed constraints. A simplification is made here to define

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the moles of en in the two sub-systems based on the assumption of the least mass change. The change of

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moles of en in (A) (and similarly in (B)) is computed from the following relation:

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C ∆nA en = ∆nen

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0 nA en

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C A0 B0 where ∆nC en = nen − nen − nen is the total change of nen in the whole system (C). With this additional

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constraint the minimization procedure has a unique solution and a complete description of the mineralogy of

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(A) and (B) can be found (Table 4). The values for the extent of the reactions are ξ1 =-0.059, ξ2 =0.060. The

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main observation of this case is that even when the initial two assemblages separately at equilibrium include

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the same mineral phases, after they are put together, distinct mineral abundances and bulk compositions

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are retained.

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2.4. case 4: peridotite-dunite, system N a2 O − CaO − M gO − F eO − Al2 O3 − SiO2 , two assemblages with

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partially different mineralogy

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The two assemblages considered in this section are a peridotite (A0 ) and a dunite (B0 ) in a compositional

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space that includes also FeO. The dunite bulk composition is obtained from a thermodynamic computation

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at P=30 kbar, T=1760o C in which extensive melting of a peridotite (A0 ) leaves an extremely depleted

256

residual solid consisting only of opx and ol. Bulk compositions and equilibrium mineral assemblages for

257

(A0 ), (B0 ) and the combined system (C) are reported in Table 5. The stoichiometric analysis determines

258

5 possible reactions: [1] di + f s ↔ hd + en, [2] f a + 2 di ↔ 2 hd + f o, [3] di + py ↔ cats + 4 en, [4]

259

alm + 4 di ↔ 3 hd + cats + 4 en and [5] grs + py ↔ cats + 4 en. The procedure applied in the previous

260

sections 2.2 and 2.3 is similarly used here. Since cpx and gt are present in (C) but not in the dunite 9

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assemblage (B0 ), it is assumed that if cpx or gt would form, more likely it would happen on the peridotite

262

side, therefore in the numerical procedure changes of cpx and gt are set to zero on the dunite side. The

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problem however is still not sufficiently constrained unless one component in either opx (en, fs) or in ol (fo, fa)

264

is determined independently. The least mass change approach introduced in the previous section (equation

265

7) is applied to fix a single component in opx or ol in (A0 ) and (B0 ). There are in total 4 components,

266

therefore 4 independent set of compositions describing (A) and (B) can be computed. All 4 set of solutions

267

are acceptable since they satisfy all the constraints. Among these 4 possibilities, the chosen compositional

268

solution for (A) and (B) is the one that requires the smaller cumulative change of opx and ol (i.e. smaller

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A A A B B B B |∆nA en | + |∆nf s | + |∆nf o | + |∆nf a | + |∆nen | + |∆nf s | + |∆nf o | + |∆nf a |). The smaller cumulative change of

270

opx and ol is found when the least mass change relation, like the one given by equation 7, is applied to fix

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the en component. Table 5 reports the final composition of (A) and (B) after equilibration. A very similar

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compositional result (less than ∼5% difference) is obtained by fixing fo instead of en. The complete result

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includes the parameters defining the extent of the reactions, ξ1 =0.007, ξ2 =0.026, ξ3 =0.016, ξ4 =-0.017 and

274

ξ5 =0.001.

275

3. Application: melting of an heterogeneous mantle in complete equilibrium.

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A better understanding of chemical equilibration in the mantle would require a further step involving

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the study of the timescale of the kinetic processes that control the chemical exchange between lithologies.

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Nevertheless it is useful to consider a limiting scenario in which an heterogeneous mantle is in complete

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chemical equilibrium. In this section are discussed preliminary results on the implications of this assumption

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for melting in the upper mantle.

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3.1. Thermodynamic melting model of two peridotites in chemical equilibrium

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The thermodynamic database applied thorough this study includes also the thermodynamic properties

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of melt in simplified chemical systems, therefore it is conveniently applied in this and the following section.

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It was shown in a previous study (Tirone et al., 2012) that the database overall is capable of reproduc-

285

ing at least approximately petrological and compositional variations consistent with our understanding of

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peridotite melting in simplified chemical systems. Given a certain bulk composition for a peridotite {1}

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(N a2 O=0.41, CaO=3.15, M gO=38.40, F eO=8.30, Al2 O3 =4.54 and SiO2 =45.20 in wt%), a simple Gibbs

288

free energy minimization at a given pressure (10 kbar) and various temperatures illustrates the variation of

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the composition of the residual solid and melt assuming a batch melting process (Fig. 4). A similar ther-

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modynamic computation is also performed on a different peridotite that is in chemical equilibrium with the

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previous one. Since the composition of the two assemblages before they have been in equilibrium together

292

is unknown, the only constrain that need to be imposed to establish chemical equilibrium at certain P,T 10

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conditions is that the chemical potential of the same mineral components must be equal for both sub-systems

294

(equation 4). As mentioned earlier, the thermodynamic database used in this study assumes ideal mixing

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for the solid solutions, which means that equality of the molar fraction of the same component in the two

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peridotites (equation 5) is also a sufficient condition for equilibration. The bulk composition defined above

297

for peridotite {1} is used to compute the equilibrium assemblage at some arbitrary P,T conditions (P=31

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kbar, T=1450o C). The assemblage consists of cpx (1.775 mol, 10.36 wt%), opx (5.396 mol, 15.12 wt%), ol

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(14.595 mol, 59.07 wt%) and gt (1.347 mol, 15.45 wt%) (solid lines in Fig. 4). The molar fraction of the

300

mineral components are: (cpx) di=0.670, hd=0.034, cats=0.023, jd=0.273, (opx) en=0.930, fs=0.067, (ol)

301

fo=0.881, fa=0.118, (gt) py=0.728, alm=0.082, grs=0.189. To generate a new assemblage {2} in equilibrium

302

with {1} at the same P,T conditions, the molar fraction of the components must be held fixed, in accordance

303

with the equilibrium constrain, while amount of each mineral can be arbitrarily modified. The change to the

304

amount of each mineral in wt % is imposed as follow: -50% cpx, +10% opx, +5% ol -50% gt. A new bulk

305

composition for peridotite {2} is then calculated from the new mineral abundance defined by the minerals

306

amount and the fraction of the components (N a2 O=0.22, CaO=1.72, M gO=41.6, F eO=9.10, Al2 O3 =2.48,

307

and SiO2 =44.87 in wt%). A thermodynamic computation using this new bulk composition at the same

308

P,T given above (P=31 kbar, T=1450o C), determines an assemblage {2} in which the component fractions

309

for each mineral are the same as in peridotite {1} but the amount of minerals is different (dashed lines

310

in Fig. 4). The batch melting thermodynamic computation applied to this new peridotite shows that the

311

composition of the residual solid is relatively similar to the composition obtained from melting assemblage

312

{1} (Fig. 4), and the difference in the composition of the two melts that are formed at the same temperature

313

is very small. This observation combined with the fact that the pattern of chemical variations as a function

314

of temperature for the two assemblages is very similar, suggest that a melting process involving two different

315

but in equilibrium peridotites should produce similar results.

316

3.2. Coupled geodynamic and thermodynamic melting model applied to an equilibrated heterogeneous mantle

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Melting of an heterogeneous mantle has been studied using mantle geodynamic numerical models in

318

combination with parameterized functions to describe the compositional variations (Ito and Mahoney, 2005;

319

Balmer et al., 2009, 2011). Recently a two-phase geodynamic melting model was applied to investigate mid-

320

ocean ridges melting of two different ideal lithologies (Katz and Weatherley, 2012; Weatherley and Katz,

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2012). The former studies focused on the melt composition and melt production while the latter put the

322

main attention on the melt distribution and melt flow. In this section are presented preliminary results on

323

the modeling of melting an heterogeneous mantle that is assumed to be in complete chemical equilibrium

324

according to the thermodynamic considerations outlined in the previous sections. The numerical details of

325

the two-phase transport model for melt and solid and the coupling with thermodynamics have been discussed

326

elsewhere (Tirone et al., 2009, 2012). Only certain aspects of the model along with the specific details related

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to the creation of chemical heterogeneities in the mantle are briefly illustrated in the following. The 2-D

328

model simulates half-space passive upwelling of the mantle with horizontal spreading velocity at the surface

329

set to 4 cm/yr. The initial composition is homogeneous (N a2 O=0.41, CaO=3.15, M gO=38.40, F eO=8.30,

330

Al2 O3 =4.54 and SiO2 =45.20 in wt%) and the temperature is equal to 1100o C everywhere in the simulation

331

model at the initial time. Bottom temperature is fixed at 1450o C and surface temperature is 25o C. The

332

permeability constant (Tirone et al., 2009) is set to 5×109 m2 . Viscosity of melt is 1 Pa s and mantle

333

viscosity is described by a temperature dependent function 8.0514 × 109 e3E5/(8.314(T (K)+273)) [Pa s]. During

334

the simulation chemical heterogeneities are introduced at the base of the model. Within a random spatial

335

interval (∆ x=500-3000 m), a bulk composition is randomly generated by varying the amount of minerals

336

in the assemblage but preserving the molar proportion of the chemical components at equilibrium at the

337

bottom of the model. This is the same equilibrium condition that was applied in section 3.1. The amount

338

of minerals is randomly varied within the following arbitrary range: ± 20 wt%, ± 15 wt%, ± 10 wt%, and

339

± 20 wt% for cpx, opx ol, and gt respectively. The heterogeneities are maintained fixed at the base of the

340

model for 50 kyr, then new random bulk compositions are created. The definition of the spatial dimension

341

of the chemical heterogeneities and the range of mineral variations are somehow arbitrary since we have

342

little or no information to make a better assessment. Fig. ?? (left panel) shows some of the initial results

343

of this model after ∼5 Myr. Olivine distribution in the solid is illustrated by the grayscale/color contours

344

on the left panel. Melting starts at ∼ 90 km depth. At greater depths variation of the amount of olivine

345

reflects only the heterogeneous mantle composition introduced at the base of the model. Above this depth

346

the abundance of olivine is the result of phase transitions and continuous melting. It can be noticed that

347

the melt distribution, described by contour lines, is not particularly sensitive to variations of olivine and

348

other mineral phases in the solid. The cold lithospheric mantle moving away from the ridge is approximately

349

defined by the 1200o C isotherm (Fig. ??). The right panels of Fig. ?? show the content of N a2 O in the

350

solid and melt at various depths. Heterogeneity of the mantle can be clearly observed from the irregular

351

distribution of N a2 O at 99 km depth. However, as melt continues to form moving upwards, the residual solid

352

and the melt show a progressively more homogeneous composition. The general but preliminary conclusion

353

is that dynamic melting of an heterogeneous mantle in chemical equilibrium does not create distinctive melt

354

compositions, in fact the residual solids tends to become rather compositionally homogeneous even when

355

some mineralogical differences still persist.

356

In Fig. ?? the results of a different model are also included in the lower three panels which show N a2 O

357

in melt at 40, 50 and 68 km depth. In the new model the heterogeneities introduced at the base of the

358

simulation model are created using a different approach. Instead of randomly varying the amount of mineral

359

phases keeping the component fractions fixed, the oxide abundance is randomly varied within the following

360

range, ± 40 wt%, ± 30 wt%, ± 10 wt%, ± 10 wt%, and ± 10 wt% for N a2 O, CaO, M gO, F eO, Al2 O3 ,

361

respectively. The wt % of SiO2 is just the remaining amount to close the sum to 100. In this way no

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equilibrium constraint is imposed between different lithologies. The setup conditions of this new model are

363

slightly different from the one presented earlier, the permeability constant is set to 1×109 m2 , the random

364

spatial interval is ∆ x=500-4000 m and new heterogeneities are created every 100 kyr. Putting aside

365

these differences, it appears that the heterogeneities in the melt source determine greater melt variations in

366

comparison to the case in which the heterogeneous mantle source is initially in equilibrium.

367

4. Conclusions

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This study begins to address the problem of understanding chemical heterogeneities in the mantle using

369

a forward approach which involves the development a quantitative description of the problem. The first

370

step discussed in this study, aims to evaluate the equilibration between two lithologies using thermodynamic

371

principles and some reasonable assumptions. The method is applied to assemblages relevant to mantle

372

problems focusing at this stage on the upper mantle. The general observation is that from a thermodynamic

373

point of view, chemical equilibration between two lithologies does not imply the formation of an homogeneous

374

distribution of minerals or uniform bulk composition, instead mineral abundance and bulk compositions

375

remain different although not the same they were before the two lithologies have been brought together.

376

This is a potentially permanent petrological condition that can be modified only by processes like melting,

377

fluid-rock interaction or mechanical mixing.

378

The next step in future to comprehend the chemical and petrological evolution of the mantle from a model

379

perspective is the determination of the time scale of the equilibration process which can only be quantified by

380

kinetic experimental studies. The kinetics of the chemical processes involved controls not only the time scale

381

but also the spatial extent of equilibration. For example if on a given time ‘t’, equilibration is established

382

over a spatial scale ‘z’, then the method outlined in this study can be applied by combining the information

383

of the two initial lithologies assembled in proportion ‘x:y’. A certain amount of experimental work has been

384

done on specific chemical transformations relevant to the mantle (e.g. Rubie and Ross II, 1994; Gard´es et

385

al., 2011; Nishi et al., 2011; Dobson and Mariani, 2014), however the results of these studies are difficult to

386

apply to complex kinetic processes involving multiple reactions in multiphase and multicomponent systems.

387

A new kinetic model and a new experimental approach are under development to specifically address the

388

extent of equilibration of two lithologies. In this new approach the pre-determination of the equilibrium

389

composition of two assemblages using the method outlined in this study is essential to build a kinetic model

390

that will allow us to quantify the time scale of chemical equilibration in the mantle.

391

In recent studies (Xu et al., 2008; Ritsema et al., 2009) geophysical computations have been performed based

392

on two end-members scenario for the petrological composition of an heterogeneous mantle. It was assumed

393

either the coexistence of two distinctive lithologies (harzburgite and eclogite) or a complete homogenization

394

of the related bulk compositions to produce an average mineralogical assemblage. The first end-member

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assumption correctly describes two lithologies in complete chemical disequilibrium. The Voigt-Reuss-Hill

396

average of the physical properties for the two lithologies used in Xu et al. (2008) and Ritsema et al. (2009)

397

seems therefore appropriate, when the two lithologies are isotropically distributed in space. The second end-

398

member case for complete equilibrium requires some clarification. In this study it was shown that complete

399

equilibration does not mean homogenization of the mineral distribution, i.e. two different assemblages

400

(A) and (B) still coexist and they are different from (C). Therefore the distribution in space of the two

401

equilibrated lithologies determines the type of averaging scheme that should be applied. In the case of

402

random/isotropic distribution in space of the two assemblages in equilibrium then Voigt-Reuss-Hill average

403

could be applied, as it was done for the first end-member case. If the two lithologies are not randomly

404

distributed then depending on their spatial distribution, a different averaging scheme should be considered

405

instead (Karato, 2008) or averaging may not be even the best option at all. The scale of the problem has an

406

influence on the determination of the phyiscal properties as well. For instance considering an asthenosperic

407

mantle bounded by subducting slab (e.g. pacific plate), an average viscosity that is assumed to vary only

408

with depth has probably little meaning even in the unlikely case that the slab is in complete chemical

409

equilibrium with the rest of the mantle. However at the CMB (for instance within the ULVZ domain), if

410

chemical heterogeneities in equilibrium or in disequilibrium are developed over a scale much smaller than

411

the extension of the plate, then it seem quite reasonable to assume an average viscosity in this domain.

412

The preliminary application of the model developed in this study for melting of two peridotite assemblages in

413

chemical equilibrium shows that the chemical evolution of the two melt products is very similar. A dynamic

414

melting model in which the mantle composition includes random chemical variations defining different

415

peridotites that are in equilibrium, reveals that the melt composition is not greatly affected by such variations

416

and the bulk composition of the residual solid tends to become quite homogeneous, essentially erasing

417

any previous compositional variations. This may provide an alternative explanation to the rather uniform

418

composition of major elements in MORBs (Macdougall, 1988; Langmuir et al., 1992; Rubin and Sinton, 2007;

419

White and Klein, 2014) which, in the common view, is described as the result of melting a compositionally

420

homogeneous mantle. More work is still needed to model melting of very different end-member lithologies

421

in chemical equilibrium (e.g. peridotie-dunite, lherzolite-eclogite). In addition a description of dynamic

422

melting that does not include the kinetic of the melt-solid interaction carries a level of uncertainty not yet

423

quantified.

424

5. Acknowledgments

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This work benefited from discussions developed over several years between one of the author (M.T.)

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and J. Ganguly. Clarity of some of the concepts presented in this study greatly improved thanks to the

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feedback from the students of a graduate course in thermodynamics and geodynamics that has been offered 14

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Many thanks to Maxim Ballmer and an anonymous reviewer for their constructive comments.

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Tackley, P., Xie, S., 2002. The thermochemical structure and evolution of Earth’s mantle: constraints and numerical models. Philosophical Transactions of the Royal Society of London A 360, 2593-2609. Tirone, M., Ganguly, J., Morgan, G.P., 2009. Modeling petrological geodynamics in the Earth’s mantle. Geochemistry, Geo-

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DOI: 10.1029/2006GL026868.

Schubert, G., Turcotte, D. L., Olson, P., 2001. Mantle convection in the Earth and planets. 1st ed., 940 pp., Cambridge

physics, Geosystems 10 doi: 10.1029/2008GC002168. Tirone, M., Sen, G., Morgan, G.P., 2012. Petrological geodynamic modeling of mid-ocean ridges. Physics of the Earth and Planetary Interiors 190-191, 51-70.

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526

Tolstikhin, I.N., Kramer, J.D., Hofmann, A.W., 2006. A chemical Earth model with whole mantle convection: The importance of a core-mantle boundary layer (DW) and its early formation. Chemical Geology 226, 79-99.

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Trampert, J, Deschamps, F., Resovsky, J., Yuen, D., 2004. Probabilistic Tomography Maps Chemical Heterogeneities Throughout the Lower Mantle. Science 306, 853-856. van Keken P.E., Hauri, E.H., Ballentine, C.J., 2002. Mantle mixing: The Generation, Preservation, and Destruction of Chemical Heterogeneity. Annual Review of Earth and Planetary Sciences 30, 493-525. van der Hilst, R.D., Widlyantoro, S., Engdahl, E.R., 1997. Evidence for deep mantle circulation fom global tomography. Nature 386, 578-584.

van Keken P. E., Ballentine, C.J., 1998. Whole-mantle versus layered mantle convection and the role of a high-viscosity lower mantle in terrestrial volatile evolution. Earth and Planetary Science Letters 156, 19-32. van Zeggeren, F., Storey, S.H., 1970. The computation of chemical equilibrium. 1st ed., 176 pp., Cambridge University Press, UK. Walzer, U., Hendel, R., 1999. A new convection-fractionation model for the evolution of the principal geochemical reservoirs of the Earth’s mantle. Physics of the Earth and Planetary Interiors 112, 211-256. Weatherley, S.M., Katz, R.F., 2012. Melting and channelized magmatic flow in chemically heterogeneous, upwelling mantle. Geochemistry, Geophysics, Geosystems 13, doi: 10.1029/2011GC003989. White, W.M., Klein, E.M., 2014. Composition of the Oceanic Crust. In: The crust. (ed. Rudnick R.L.). Treatise on Geochemistry (Second Edition), volume 4. Elsevier. Xu, Y.G., 2001. Thermo-Tectonic Destruction of the Archaean Lithospheric Keel Beneath the Sino-Korean Craton in China: Evidence, Timing and Mechanism. Physics and Chemistry of the Earth (A) 26, 747-757 Xu, W., Lithgow-Bertelloni, C., Stixrude, L., Ritsema, J., 2008. The effect of bulk composition and temperature on mantle

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UK.

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and upper mantle temperature. Journal of Geophysical Research 111, doi: 10.1029/2005JB003972. Zhu, C., Anderson, G., 2002. Envoromental applications of geochemical modeling. 1st ed., 284 pp., Cambridge University Press,

MA

560

D

559

seismic structure. Earth and Planetary Science Letters 275, 70-79. Zhong, S., 2006. Constraints on thermochemical convection of the mantle from plume heat flux, plume excess temperature,

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557

18

ACCEPTED MANUSCRIPT

&l*

&m*

,+ &'( )*

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4 547876 54789 8 6 5 78:6 549876 549896 5498:6 ;;; - &'( )* . %+ / ,+ , &'( )*

123 @A1BC 5D787F L 5D789F L 5D78:F L 5D 987F L 5D 989F L 5D 98:F L ;;;

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<=>? @A1BC 5K4GH F L 5KIJH F L ;;;

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- &'( )* . %+ / ,+ , &'( )*

% &'( )*

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123 @A1BC <=>? @A1BC 5KE4GH F ;;; 5KE787F ;;; 5KEIJH F ;;; 5KE789F ;;; 5KE78:F ;;; 0 123 ;;; 5KE987F ;;; 5KE989F ;;; 5KE98:F ;;; ;;;

%+ &'( )*

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%+ &'( )* <=>? @A1BC 123 @A1BC 5DE4GH F ;;; 5DE787F ;;; DE 5 IJH F ;;; 5DE789F ;;; ;;; 0 123 5DE78:F ;;; M NOP QR F SE T UE 5DE987F ;;; ZVV]^ [ ZWXV]^ \ ZYXV]^ Z V_`^ [ ZWX_`^ \ ZYX_`^ 5DE989F ;;; aaa 5DE98:F ;;; 0 123 ;;;

bc5defg hifjdjk

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#

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D

Figure 1: Illustration of the problem in this study. Assemblage (A0 ) and (B0 ) are separately in thermodynamic equilibrium at

TE

given P,T conditions. Bulk composition is known and mineral composition is computed from the equilibrium thermodynamic calculation. Once the two assemblages are together at the same P,T conditions in a certain proportion (in the grey box the proportion is 1:1), bulk composition of the new system (C) is easily obtained as well as minerals abundance and composition

AC CE P

from equilibrium thermodynamic calculation. Finding the bulk and minerals composition in the two sub-systems (A) and (B) is the objective of this study. On the right panels it is shown an hypothetical example in which after equilibration the amount of mineral 1 decreases and the amount of mineral 3 increases on the left sub-system (A), mineral 3 increases and mineral 5 is completely exhausted on the right sub-system (B). Note that the right lower panel does not describe the exact volume of the two sub-systems at equilibrium or the exact mineral spatial distribution in (A) and (B). Bulk composition is defined by the moles of the oxides nM gO , nCaO ,.. Mineral composition for phase i is defined by the moles of the components ni1 , ni2 ,..

19

ACCEPTED MANUSCRIPT

20

RI

PT

Tirone et al. / Lithos 00 (2015) 1–??

0.02 B

-0.01

A

A

∆ G /G0

SC NU

0

MA

∆ G/G0

0.01

∆ G /GB0

-0.02

1.5

2

AC CE P

TE

0.001

D

∆n

2.5

A en

∆ G/G0

0.0005

0

|∆ GA/GA0 |+|∆ GB/GB0 | A

A

∆ GB/GB0

∆ G /G0

-0.0005 1.7 1.725 1.75 1.775 1.8 1.825 1.85 A

∆ nen

Figure 2. (upper panel) Case 1 section 2.1. Variation of the Gibbs energy in the peridotite and eclogite assemblages ∆GA /GA 0 = A B B B B B (GA − GA 0 )/G0 , ∆G /G0 = (G − G0 )G0 as a function of the change of moles of en on the peridotite side (change of en on A A 2 B B 2 the eclogite side is given by 2.515 − ∆nA en ). (lower panel) enlarged view. The function [∆G /G0 ] + [∆G /G0 ] is used to B B locate the distribution of en on each sides. For a more clear view, in this figure is plotted |∆GA /GA 0 | + |∆G /G0 |.

20

ACCEPTED MANUSCRIPT

21

Tirone et al. / Lithos 00 (2015) 1–??

PT

1.9

ngtB

RI

1.85

1.75

SC

ngt

1.8

ngtA

1.65

MA

1.6

NU

1.7

1.55 0

0.1

0.3

A en

D

∆n

0.2

TE

Figure 3. Case 3 section 2.3. Variation of the moles of gt in the peridotite and depleted peridotite as a function of the change of moles of en in (A). All values satisfy the imposed constraints. The chosen composition (vertical line) is the one based on

AC CE P

the assumption of the least change of en moles (see equation 7).

21

solid (10 kb) MgO

20

gt 10

0

10

-1

10

-2

10

-3

CaO Na2O

cpx

1200

D

P=31 kb

SC

10

NU

opx

MA

ol

30

FeO Al2O3

101

oxides (wt %)

40

SiO2

RI

wt % 60 50

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ACCEPTED MANUSCRIPT

1300

o

1400

temperature ( C)

TE

melt (10 kb)

wt %

AC CE P

50 40 30

SiO2 Al2O3

20

CaO FeO

10

MgO melt Na2O 1200

1300

1400

temperature (oC)

Figure 4: (upper left panel) Mineral abundance of two peridotites in chemical equilibrium at 1450o C and ∼ 31 kbar. Solid lines peridotite {1}, dashed lines peridotite {2}. (upper right panel) Bulk composition of the residual solid (wt %) during batch melting at 10 kbar. (lower panel) Bulk composition of the related liquids (wt %). The two thicker lines illustrate the melt abundance in wt %.

22

ACCEPTED MANUSCRIPT

40

60

80

0

100

0

0.5

10

0.45

5

0.35

4 3

> 62

1200oC

50 2

60 70

1 80

1

100

< 52 0.05 ol wt%

o

0 -10 -20 0

10

20 10 0

-10

20

30 km

-20 0

10

20

30 km

10

z=68 km

9

7 6 5

z=40 km

z = 50 km

11

8

z=50 km

TE

10

% difference from in solid

z = 68 km

AC CE P

% difference from in solid

T = 1450 C 20

z=68 km

MA

0.5

z=99 km

D

90

0.3 0 00.25 0 0 0.2 0 0 00.15 0 0 0.1 0

12

SC

40

km

10 20 30 40 50

13

0.4

6

0

14

% difference from in solid

30

10 20 30 40 50

NU

20

depth (km)

20

PT

0

melt km Na2O (wt%)

solid Na2O (wt%)

distance from the ridge axis (km)

RI

4 cm/yr →

20

z=50 km z=40 km

z = 40 km

10 0 -10 -20 0

10

20

30 km

Figure 5: (left panel) Half-space ridge melting model of an heterogeneous mantle in chemical equilibrium. Spreading velocity is 4 cm/yr, bottom temperature is 1450 o C. (left panel) Color contours show the olivine distribution. Red contour lines refer to the melt abundance wt % (0.5 % interval). White streamlines illustrate the direction of mantle flow. The lithospheric boundary is approximately defined by the 1200o C isotherm. (middle and right panel) N a2 O content (wt %) at 3 depths in the solid and melt respectively. The lower three panels show a comparison of N a2 O wt % difference in melt with respect to the average value near the ridge axis at three depths from the equilibrium heterogeneous model above here and another simulation in which the heterogeneities are not in equilibrium (see the main text for further explanation).

23

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Table 1: Name, abbreviation and formula of mineral phases and components. Footnote: thermodynamic properties of quarz

PT

not included in the previous version of the database (Tirone et al., 2012).

cpx

diopside

di

CaM gSi2 O6

hedenbergite

hd

CaF eSi2 O6

Ca-tschermaks

cats

CaAl2 SiO6

jadeite

jd

N aAlSi2 O6

orthopyroxene

opx

enstatite

en

M gSiO3

ferrosilite

fs

F eSiO3

olivine

ol

forsterite

fo

M g2 SiO4

fayalite

fa

F e2 SiO4

spinel

sp

M gAl2 O4

garnet

gt

pyrope

py

almandine

alm

†

MA

D

TE

M g3 Al2 Si3 O12 F e3 Al2 Si3 O12

AC CE P

quarz

†

RI

clinopyroxene

grossular

formula

SC

abbreviation

NU

name

grs

Ca3 Al2 Si3 O12

qz

SiO2

∆Hf (298 K, 1 bar)=-932627 J mol−1 , S(298 K, 1 bar)=33.207J K−1 mol−1

Cp(T,1 bar)[J K−1 mol−1 ]=80.01 − 2.403E2/T 1/2 − 35.467E5/T 2 + 49.157E7/T 3 V(P,T)[J bar−1 ]=2.37(1 − 1.238E−6(P − 1 bar) + 7.087E−12(P − 1 bar)2 )

24

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Table 2: Case 1, section 2.1, System: CaO − M gO − Al2 O3 − SiO2 . Bulk composition of two assemblages, peridotite=(A0 ), eclogite=(B0 ) and relative mineralogical compositions at equilibrium computed from two independent Gibbs free energy min-

PT

imizations. Assemblage (C) is defined by a combination of the oxide components in (A0 ) and (B0 ). Minerals composition in (A) and (B) are retrieved following the method discussed in section 2.1. The combined minerals composition of (A) and (B) is consistent with the equilibrated minerals composition obtained from a Gibbs free energy minimization applied to the bulk

bulk comp.

(A0 )

(C)†

(B0 )

wt%

mol

wt%

mol

mol

CaO

3.46

2.25

12.29

8.02

10.27

MgO

46.70

41.15

22.81

20.72

Al2 O3

4.54

3.16

15.67

11.25

SiO2

45.20

53.43

49.23

60.00

(B)

wt%

mol

wt%

mol

3.56

2.25

12.29

8.05

61.87

45.84

40.29

23.63

21.52

14.41

4.54

3.15

15.68

11.29

46.06

54.31

48.40

59.14

MA

min. comp.

(A)

NU

oxides

SC

RI

composition of (C). Bulk compositions of (A) and (B) are calculated from their respective mineral compositions.

113.43

———————————————— mol ———————————— 1.08×0.991‡

3.84×0.991

4.93×0.991

1.071¶ (0)§

3.813(0)

cats(cpx)

1.08×0.009

3.84×0.009

4.93×0.009

0.010(0)

0.034(0)

en(opx)

3.160×1

4.304×1

9.979×1

4.933(1.773)

5.046(0.742)

fo(ol)

16.689×1

0

15.432×1

15.432(-1.257)

0(0)

py(gt)

1.57×0.751

5.59×0.751

1.180(0)

4.201(0)

TE

qz

7.16×0.751

1.57×0.249

5.59×0.249

7.16×0.249

0.391(0)

1.392(0)

0

1.257×1

0

0(0)

0(-1.257)

AC CE P

grs(gt)

D

di(cpx)

†

moles(C)=moles(A)+moles(B)

‡

moles of the mineral phase×molar component fraction = moles of the mineral component

¶

moles of the mineral component

§

in brackets: change of the moles of the components (∆nij ) after equilibration of (A) and (B) together

25

ACCEPTED MANUSCRIPT

Table 3: Case 2, section 2.2, System: N a2 O − CaO − M gO − Al2 O3 − SiO2 .

Bulk composition of two assemblages,

peridotite=(A0 ), eclogite=(B0 ) and relative mineralogical compositions at equilibrium computed from two independent Gibbs

PT

free energy minimizations. Assemblage (C) is defined by a combination of the oxide components in (A0 ) and (B0 ). Minerals composition in (A) and (B) are retrieved following the method discussed in section 2.2. The combined minerals composition of (A) and (B) is consistent with the equilibrated minerals composition obtained from a Gibbs free energy minimization applied

RI

to the bulk composition of (C). Bulk compositions of (A) and (B) are calculated from their respective mineral compositions.

bulk comp.

(A0 )

(B0 )

(C)

wt%

mol

wt%

mol

mol

Na2 O

0.41

0.47

2.85

3.43

3.90

CaO

3.15

1.99

11.75

7.82

MgO

46.70

41.06

13.00

12.03

Al2 O3

4.54

3.16

25.87

18.93

SiO2

45.20

53.32

46.53

57.79

cats(cpx)

1.69×0.008

jd(cpx) en(opx)

qz

0.41

0.49

2.83

3.26

9.81

7.76

5.15

7.36

4.68

53.09

24.47

22.58

34.09

30.15

22.09

22.75

16.59

8.62

6.03

111.11

44.60

55.20

47.09

55.88

0.662(-0.546)

4.811(4.811)

11.18×0.695

9.49×0.009

0.010(-.003)

0.007(-7.698)

1.69×0.277

11.18×0.305

9.49×0.411

0.468(0)

3.407(0)

1.971×1

0

0(-1.971)

0(0)

0

17.332×1

0.227×1

13.771×1

1.055(-16.278)

12.659(6.642)

0

0

1.378×1

0(0)

1.378(1.378)

1.33×0.807

0

7.63×0.815

6.266(5.912)

0.008(0.008)

1.33×0.193

0

7.63×0.184

1.415(1.158)

0.002(0.002)

0

0.133×1

0

0(0)

0(-0.133)

AC CE P

grs(gt)

mol

9.49×0.580

D

0

py(gt)

wt%

TE

1.69×0.715

sp

mol

———————————————— mol ————————————

di(cpx)

fo(ol)

(B)

wt%

MA

min. comp.

(A)

NU

oxides

SC

Footnotes of Table 2 apply here.

26

ACCEPTED MANUSCRIPT

Table 4: Case 3, section 2.3, system: N a2 O − CaO − M gO − Al2 O3 − SiO2 .

Bulk composition of two assemblages,

peridotite=(A0 ), depleted peridotite=(B0 ) and relative mineralogical compositions at equilibrium computed from two in-

PT

dependent Gibbs free energy minimization. Assemblage (C) is defined by a combination of the oxide components in (A0 ) and (B0 ). Minerals composition in (A) and (B) are retrieved following the method discussed in section 2.3. The combined minerals composition of (A) and (B) is consistent with the equilibrated minerals composition obtained from a Gibbs free energy

RI

minimization applied to the bulk composition of (C). Bulk compositions of (A) and (B) are calculated from their respective

(A0 )

(B0 )

(C)

wt%

mol

wt%

mol

mol

Na2 O

0.41

0.47

0.42

0.58

1.05

CaO

3.15

1.99

4.65

1.05

MgO

46.70

41.06

44.7

39.45

Al2 O3

4.54

3.16

7.04

4.91

SiO2

45.20

53.32

45.60

54.00

min. comp.

(B)

wt%

mol

wt%

mol

0.41

0.47

0.51

0.58

3.04

2.18

1.38

2.62

1.66

80.51

47.21

41.51

44.42

39.01

8.07

5.34

3.71

6.27

4.36

115.39

44.86

52.93

46.16

54.39

MA

oxides

(A)

NU

bulk comp.

SC

mineral compositions. Footnotes of Table 2 apply here.

———————————————— mol ———————————— 1.69×0.715

0.93×0.364

2.44×0.560

0.608(-0.600)

0.759(0.421)

cats(cpx)

1.69×0.008

0.93×0.005

2.44×0.007

0.007(-0.005)

0.010(0.004)

jd(cpx)

1.69×0.277

0.93×0.631

2.44×0.432

0.468(0)

0.585(0)

en(opx)

1.971×1

3.996×1

6.322×1

2.088(0.117)

4.234(0.238)

py(gt) grs(gt)

TE

AC CE P

fo(ol)

D

di(cpx)

17.332×1

14.675×1

32.006×1

17.332(0)

14.675(0)

1.33×0.807

2.16×0.891

3.49×0.842

1.355(0.281)

1.584(-0.340)

1.33×0.193

2.16×0.109

3.49×0.158

0.254(-0.003)

0.297(0.063)

27

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ACCEPTED MANUSCRIPT

RI

Table 5: Case 4, section 2.4, system: N a2 O − CaO − M gO − F eO − Al2 O3 − SiO2 . Bulk composition of two assemblages, peridotite=(A0 ), dunite=(B0 ) and relative mineralogical compositions at equilibrium computed from two independent Gibbs

SC

free energy minimization. Assemblage (C) is defined by a combination of the oxide components in (A0 ) and (B0 ). Minerals composition in (A) and (B) are retrieved following the method discussed in section 2.4. The combined minerals composition of (A) and (B) is consistent with the equilibrated minerals composition obtained from a Gibbs free energy minimization applied to the bulk composition of (C). Bulk compositions of (A) and (B) are calculated from their respective mineral compositions.

(A0 )

(B0 )

(C)

wt%

mol

wt%

mol

Na2 O

0.41

0.48

0

0

CaO

3.15

2.06

0

0

MgO

38.40

34.88

47.53

FeO

8.30

4.23

7.72

Al2 O3

4.54

3.26

0

0

SiO2

45.20

55.09

44.75

53.65

di(cpx)

wt%

mol

wt%

mol

0.48

0.41

0.49

0

0

2.06

3.17

2.06

0

0

42.47

77.35

39.16

35.41

46.70

41.93

3.87

8.10

7.28

3.70

8.74

4.40

3.26

4.57

3.27

0

0

108.74

45.39

55.07

44.56

53.67

D

———————————————— mol ————————————

AC CE P

min. comp.

(B)

mol

MA

oxides

(A)

TE

bulk comp.

NU

Footnotes of Table 2 apply here.

1.76×0.686

0

1.76×0.690

1.215(0.005)

0(0)

1.76×0.032

0

1.76×0.028

0.049(-0.008)

0(0)

1.76×0.007

0

1.76×0.007

0.012(0.0001)

0(0)

1.76×0.275

0

1.76×0.275

0.484(0)

0(0)

5.31×0.931

7.31×0.948

12.62×0.940

4.939(-0.001)

6.921(-0.001)

5.31×0.069

7.31×0.052

12.62×0.060

0.315(-0.051)

0.442(0.058)

fo(ol)

14.59×0.884

19.52×0.911

34.11×0.898

13.089(0.191)

17.559(-0.217)

fa(ol)

14.59×0.116

19.52×0.089

34.11×0.102

1.481(-0.216)

1.987(0.242)

py(gt)

1.37×0.712

0

1.38×0.724

0.996(0.016)

0(0)

alm(gt)

1.37×0.100

0

1.38×0.087

0.120(-0.017)

0(0)

grs(gt)

1.37×0.188

0

1.38×0.189

0.260(0.001)

0(0)

hd(cpx) cats(cpx) jd(cpx) en(opx) fs(opx)

28

ACCEPTED MANUSCRIPT !"#$%"#$&'(

PT RI SC NU MA D TE

-

From a thermodynamic point of view, two assemblages in equilibrium preserve different mineralogy and composition indefinitely A method to compute the equilibrium composition of two lithologies in contact is presented Melt products of two different peridotites in chemical equilibrium are very similar

AC CE P

-