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Abstract

When nucleation and growth of minerals in rocks occur, the system as a whole approaches a lower energy state. At given temperature, pressure and composition, the stable state corresponds to the minimum value of the rock Gibbs function. However, a description in term of equilibrium is not appropriate to understand the genesis of spatially ordered dispositions of minerals, non-equilibrium minerals morphology, zonation, and all features preceding equilibrium. Very commonly the kinetics of the transformation is frozen in, and disequilibrium textures beautifully show up. These situations must be described in terms of flow of components driven by the gradients of chemical potentials. For instance when nucleation occurs, the components' chemical potential differences are determined by the local associations of minerals in different parts of the rock (Fisher, 1973, 1977). The relative rates of intergranular diffusion and of mineral growth and dissolution determine the steepness of these gradients (Fig. 1).

Introduction

When nucleation and growth of minerals in rocks occur, the system as a whole approaches a lower energy state. At given temperature, pressure and composition, the stable state corresponds to the minimum value of the rock Gibbs function. However, a description in term of equilibrium is not appropriate to understand the genesis of spatially ordered dispositions of minerals, non-equilibrium minerals morphology, zonation, and all features preceding equilibrium. Very commonly the kinetics of the transformation is frozen in, and disequilibrium textures beautifully show up. These situations must be described in terms of flow of components driven by the gradients of chemical potentials. For instance when nucleation occurs, the components' chemical potential differences are determined by the local associations of minerals in different parts of the rock (Fisher, 1973, 1977). The relative rates of intergranular diffusion and of mineral growth and dissolution determine the steepness of these gradients (Fig. 1).

Fig. 1.

Schematic chemical potential (&) gradients around a growing crystal for elements with very slow, slow, medium, and fast rates of intergranular diffusion. From Carlson (2002), modified.

Fig. 1.

Schematic chemical potential (&) gradients around a growing crystal for elements with very slow, slow, medium, and fast rates of intergranular diffusion. From Carlson (2002), modified.

If the rate of intergranular diffusion controls the nucleation and growth of the new phases a spatial organisation of reactant and product minerals results. Discussion on this subject can be found in Fisher (1973, 1977) and references therein, where linear non-equilibrium thermodynamics is used to tackle the situations of disequilibrium between growing minerals and matrix.

When disequilibrium prevails at thin section level, we can have access to a fragment of the metamorphic history of the rock from which the section was obtained. But what is measured is a consequence of an ensemble of kinetic processes, among which the slowest are recorded. Yet to have access to the history, models are needed in order to describe segregation of elements in minerals, inter- and intracrystalline element diffusion, growth and dissolution rates. The models are built from experiments and physical laws. If we had a complete model we could fully exploit the information from a thin section, building a bridge from measured quantities to pressure, temperature and time.

The ideal situation is not the actual one and different approaches are used, taylored on the problem at hand. Some approaches couple a petrographical description of UHP rocks to petrogenetic grids, but the kinetics is somewhat overlooked. We choose to describe a few kinetic models in some detail, because they can be suitable for an analysis of rock textures formed in a great interval of temperature and pressure values. The examples refer to relatively simple textures allowing a more didactic presentation. So this chapter aims a) to illustrate some models proposed to explain the formation of textures frequently observed in rocks, including the UHP metamorphic ones; b) to report two applications of the diffusion equation to retrive relevant information from zoned minerals; c) to show how equilibrium thermodynamics can help i) understanding disequilibrium features at thin section scale and ii) to recognise ultrahigh pressure mineral associations; this can be a delicate task when associations are stable over a wide interval of pressure values.

Kinetic theory

Nowadays the study of the rate of mineral transformations has a great importance in earth sciences. It lies at the intersection of many fields: chemical kinetics, transport theory, nucleation and crystal growth, structural phase transitions, dislocation theory, etc. and it benefits of the progress of the knowledge of the behaviour of organic and inorganic materials which are not minerals. A powerful phenomenological description of the processes involved in mineral transformation is furnished by the linear thermodynamics of irreversible processes. It also gives an understanding of diffusion deeper than Fick's first and second laws alone. Moreover it can describe the coupling between transport phenomena such as thermal diffusion, where a concentration gradient sets in an initially homegeneous solution due to a temperature gradient. The literature on irreversible thermodynamics is vast: several references (De Groot, 1951; De Groot & Mazur, 1962; Fitts, 1962; Katchalsky & Curran, 1965; Prigogine, 1968; Haase, 1969; Kaiken, 1994) are suggested to the interested readers. Some notions of irreversible thermodynamics are recalled in Appendix because several subjects reviewed in this chapter are based on it.

Growth of mineral layers

A typical situation often observed is the metasomatic reaction of two-mineral assemblages which produce a layered sequence of reaction products, sometimes arranged in concentric shells. The treatment of metasomatic processes dates back to the early works by Korzhinskii (1959), Thompson (1959), Fisher (1973, 1977). A review was made by Rubie (1990). We follow the treatment and notation of Joensten (1977) and its subsequent evolution due to Nishiyama (1983) and Johnson & Carlson (1990). A rich bibliography can be found in the paper by Joensten (1977); from this paper we take the following model of the genesis of a reaction band.

The formation of mineral layers is often associated to complex texture, owing to the sluggishness of transport processes, such as the formation of kelyphite and symplectite (Messiga & Bettini, 1990). The model we are going to illustrate revealed useful insights in these complex cases as well.

The case studied has been selected for its simplicity and because it clearly illustrates the meaning of local equilibrium and diffusion limited growth. Although the example refers to low pressure rocks the model is applicable when (constant) temperature and pressure determine diffusion rates lower than rates of mineral nucleation and growth/ dissolution and ordered sequences of mineral layers are observed. However if the minerals within a layer are zoned or some reactions are sluggish this model becomes inaccurate.

Let us consider calcite CaCO3 adjacent to anorthite CaAl2Si2O8. Reaction bands can develop as a consequence of increasing temperature. A reaction band is a layer of product minerals separating the reactant ones, as symbolically drawn in the block diagram of Figure 2. The mineral facies in the four components system CaO - AlO15 - SiO2 - CO2 are seen in Figure 2. When temperature increases calcite (C) and anorthite (A) are incompatible and react with one another (Fig. 2).

Fig. 2.

Stable minerals in the system CaO - AlO15 - SiO2 - CO2: The minerals in (b) are stable at higher temperature and lower pressure than in (a). Block diagram in (b) shows the sequence of mineral layers that results from reaction between blocks of calcite and anorthite placed in contact in (a). From Joensten (1977), modified.

Fig. 2.

Stable minerals in the system CaO - AlO15 - SiO2 - CO2: The minerals in (b) are stable at higher temperature and lower pressure than in (a). Block diagram in (b) shows the sequence of mineral layers that results from reaction between blocks of calcite and anorthite placed in contact in (a). From Joensten (1977), modified.

The assemblage C-A will initially lie at some point on the metastable line, intersection of the two potential surfaces. If the kinetics is controlled by grain boundary diffusion a reaction band is produced. Once CaSiO3 wollastonite (W) and Ca2Al2SiO7 gehlenite (G) nucleate, there may be a transient of reaction controlled growth. But growth becomes diffusion controlled as soon as W and G become large enough. The dissolution-precipitation kinetics being fast in comparison to diffusion, local equilibrium among minerals and intergranular fluid sets in. Chemical potentials, in the grain boundary where three phases meet, will shift to the invariant point. Chemical potential gradients, pinned between the two three phases invariant points (C| W + G and W + G |A), exist across the band of product minerals. The calculated values (Joensten, 1977) of the chemical potential differences of the mobile components, at the invariant associations are (Fig. 3):

Fig. 3.

Projection of saturation surface along µCaO axis, onto µAlO1µSio2 plane. Chemical potentials for 1027 °C and 350 bars. From Joensten (1977), modified.

Fig. 3.

Projection of saturation surface along µCaO axis, onto µAlO1µSio2 plane. Chemical potentials for 1027 °C and 350 bars. From Joensten (1977), modified.

 

formula

In response to the gradients, mobile components diffuse between the reactant phases. Summarising, following Johnson & Carlson (1990), the Joensten assumptions are:

  • the system evolves close to equilibrium so that the linear phenomenological laws apply between driving forces and fluxes;

  • the local equilibrium between diffusing components and adjacent minerals is set in: then the Gibbs-Duhem equation for every mineral constrains the components' chemical potentials in the solids and in the intergranular medium through the equilibrium condition µimineral = µimedium;

  • the system is set in a state of minimal rate of entropy production so that component concentrations are time independent and the flow of matter by diffusion and the reaction rates are exactly balanced (De Groot & Mazur, 1962), in this case at each reaction front;

  • chemical reactions occur only at the layer contacts but not within layers;

  • diffusion occurs within the mineral layers but non within the minerals;

  • the metamorphism is isochemical.

The isochemical constraint is not essential to the model and in Johnson & Carlson (1990) it is removed. The isochemical steady-state diffusion model requires simultaneous solution of a set of four types of linear equations describing reactions and fluxes in the growing corona. The unknowns are c + p : c are the stoichiometric coefficients of system components and p those of the reacting phases (phase components) at every interface. The diffusing components are related, through the boundary conditions of the steady state diffusion equations, to the fluxes, in the grain boundary, within the corona (or reaction band). In the case under consideration, at the interface C | W + G there are 3 phases, four components, three of them are mobile.

  • Mass-balance equations are required for each chemical component at every interface. In the case we are considering, the interfaces, indicated by a vertical bar (|), are: C | W + G |A. The diffusing components are: CaO, AlO15, SiO2 while it is supposed that no CO2 gradients exist. CO2 is evolved at the boundary of the reaction band with calcite and its concentration becomes homogeneous after a transient which is short, compared with the formation time of the banded structure. 
    formula
    (1)
     
    formula
    (2)
    where  
    formula
    is the number of mole of component i consumed (if it is negative), or evolved in the reaction at the contact  
    formula
    is the amount in moles of phase ϕ involved in the reaction and  
    formula
    the stoichiometric coefficient i in the formula of phase ϕ.So for instance, for the component CaO, the equation at the boundary C|W + G reads:  
    formula
    and at the boundary  
    formula
    The opposite sign in the equations is a consequence of the closed system assumption  
    formula
    . There are 4 equations at the boundary C | W + G and 3 at the boundary W + G|A, in this example.
  • Steady diffusion equations
    • - Within the layers no chemical reaction occurs and

      graphic
      is constant (Nishiyama, 1983).

    • - The amount of each component entering into or leaving the intergranular medium at each interface depends only on the fluxes of that component on either side of the interface:

     
    formula
    (3)
     
    formula
    (4)
    The phases C and A are homogeneous and components do not enter or escape outside the reaction fronts so it is graphic If the gradient of chemical potential of component i is negative in going from calcite to anorthite, that component enters the corona W + G at the interface C|W + G and it leaves the corona at W + G|A. It is considered JCO2 = 0 everywhere.These conditions are in fact boundary conditions for the flux in the mineral layers and some comments are needed. It is assumed that the layers growth rate is negligible and the mass conservation equations are written in a frame fixed on the layer contact C|W + G. At the steady state the mass conservation of component k in a volume around the interface C| W + G is (Nishiyama, 1983): 
    formula
    (5)
    Equation 5 is obtained integrating, over a unit volume, the equation of conservation of matter with the term  
    formula
    corresponding to the production of component k by chemical reaction. After a further integration of Equation 5, on a unit time, we obtain: 
    formula
    (6)
    Defining a reference state such that ξ = 1 (see last point) the conditions (3) and (4) are obtained.
  • Flux ratio equations. Within the layer W + G the fluxes are related to the chemical potential gradients by: 
    formula
    (7)
    Note that cross coupling coefficients (Lij = 0) are neglected and Lii is assumed independent on composition for all components. Only c - 1 fluxes are independent (∑ J = 0), so there are c - 1 equations. Because the individual Lii are not known it is usual to write flux ratio equations (in the following JSWOG is the common divisor): 
    formula
    (8)
     
    formula
    (9)
    • -Gibbs-Duhem equations. These are p equations, one for every phase in the reaction band, relating the chemical potential of the c components. The general form is
      graphic
      . For the reaction band W + G: 
      formula
      (10)
       
      formula
      (11)
      The equilibrium betweeen minerals and intergranular fluid requires:Because we are interested in c - 1 ratios of chemical potential gradients and there are p equations among them, c - 1 -p are the free ratios. Out of the c - 1 flux ratio equations c - 1 - (c - 1 - p) are independent. So there are p = 2 independent flux ratio equations in the band W + G.
  • one extent of reaction is fixed. In conclusion there are c + p unknowns and c - 1 + p equations among stoichiometric coefficients of system components and of phase components, because only c - 1 fluxes are independent. A coefficient must be fixed by one reaction extent at one boundary. In more general cases one extent-of-reaction equation is needed for each equilibrium boundary that has been overstepped before reaction begins (Johnson & Carlson, 1990). In the case we are dealing with, a single reaction has been overstepped.The following equations describe the reaction at the interface C|W + G: 
    formula
    (14)
     
    formula
    (15)
     
    formula
    (16)
     
    formula
    (17)
     
    formula
    (18)
     
    formula
    (19)
     
    formula
    (20)
    One extent of reaction is fixed: 
    formula
    (21)
    The steady diffusion equations lead to: 
    formula
    (22)
     
    formula
    (23)
     
    formula
    (24)
    The following equations describe the reaction at the interface W + G| A: 
    formula
    (25)
     
    formula
    (26)
     
    formula
    (27)
    Indicating by M, X, B the matrices of the system coefficients, of the unknowns and known terms, the equations can be written in condensed matrix form MX = B: 
    formula
    (28)
    After substituting in the appropriate equations the values
    graphic
    obtained from the system (10-11), and for the tentatively assigned values
    graphic
    the solution (29) of the linear system (28) allows to write down the reactions (Joensten, 1997):
  • at the boundary C|W + G: 
    formula
  • at the boundary W + G| A: 
    formula
    The components produced in the reaction at C|W + G are quantitatively consumed at the boundary W + G| A and vice-versa.

 

formula
(29)

The cycle is reported in Figure 4. With the assumed values of the ratios of the phenomenological coefficients, wollastonite and gehlenite are produced at both contacts and then the reaction band W + G is stable. With different values of these ratios other sequences of reaction bands are stable. The modal proportion of wollastonite and gehlenite produced by reaction at each contact is different and changes in a stepwise manner across the layer: the observation of this steep variation is a check of the constancy of the flux ratios. A conclusive remark: the diffusion controlled reactions at the boundary of the layers are nicely tuned and make a closed reaction cycle. The observed products within the reaction band indicate that the gradient of CO2 is absent. In the contrary case we should observe assemblages reflecting the variations of the chemical potential of carbon dioxide along the reaction band. Therefore, in the Joensten model the diffusional control is exerted by fewer components than required to describe the thermodynamic equilibrium as remarked by Johnson & Carlson (1990) who explicitly calculated the number of components controlling diffusion as a function of the observable structure of the reaction layers. If minerals are mixtures of end-members, when the rate of a reaction of the cycle changes with temperature and/or composition, not only the products' modal distribution is affected but also their composition. In principle one should tackle the full diffusion-reaction problem in the network of the grain boundary, as well as in the growing and resorbing phases. The constraints on element migration will be used in the following section to model this problem in an approximate way.

Fig. 4.

Closed cycle of diffusion controlled reactions exchanging CaO, AlO15 and SiO2 between the isothermal-isobaric invariant assemblages C + W + G and W+ G + A, resulting in growth of the W + G layer at the expense of C and A. From Joensten (1977), modified.

Fig. 4.

Closed cycle of diffusion controlled reactions exchanging CaO, AlO15 and SiO2 between the isothermal-isobaric invariant assemblages C + W + G and W+ G + A, resulting in growth of the W + G layer at the expense of C and A. From Joensten (1977), modified.

Symplectic reaction in olivine

As a second example of applications of the thermodynamics of irreversible processes, we propose the symplectic reaction in olivine studied by Ashworth & Chambers (2000). To our knowledge, it is the only study dealing quantitatively with these structures. It points out that symplectites form when a modulated flux sets in a direction parallel to the growth front. Examples of symplectite after eclogite are rewieved by Rubie (1990).

The symplectites studied (Figs. 5-7) are from the Lilloise intrusion. The Lilloise is an 8 X 4 km layered mafic intrusion which cuts the plateau basalts of the East Greenland Tertiary province (Chambers & Brown, 1995). Lilloise was intruded at about 50 Ma, 4-5 Ma after cessation of the voluminous tholeiitic magmatism which accompanied the rifting of the East Greenland continental margin. The layered intrusion can be divided into three zones, the Lower, Middle and Upper, consisting of basal olivine-clinopyroxene cumulates (600 m) passing upwards into gabbroic olivine-clinopyroxene-plagioclase cumulates (1800 m) and then into a series of plagioclase-amphibole cumulates (400 m). The symplectites are found in olivine-chromo spinel cumulates (Lower zone) as thin platelets of clinopyroxene and magnetite parallel to (100) of olivine when it is less ferroan than Fo74. The symplectites are concentrated near the centres of unzoned olivine grains and tend to be associated with subgrain boundaries. For these reasons the authors think that the symplectites grew after some deformation, and in those grains where minor components were unable to diffuse out of the olivine grains. The symplectite growth is, therefore, referred to exsolution of minor components from olivine. The overall reaction can be represented by:

Fig. 5.

Rod symplectites in troctolite from Belhelvie, northeast Scotland. Hornblende-spinel symplectite in which spinel (lighter) is the rod mineral. From Ashworth & Chambers (2000), modified.

Fig. 5.

Rod symplectites in troctolite from Belhelvie, northeast Scotland. Hornblende-spinel symplectite in which spinel (lighter) is the rod mineral. From Ashworth & Chambers (2000), modified.

Fig. 6.

Symplectite of clinopyroxene and magnetite in Lilloise olivine. Optical micrographs, plane polarised light. Thin section cut parallel to (010) so that the symplectite platelets are viewed edgewise. From Ashworth & Chambers (2000), modified.

Fig. 6.

Symplectite of clinopyroxene and magnetite in Lilloise olivine. Optical micrographs, plane polarised light. Thin section cut parallel to (010) so that the symplectite platelets are viewed edgewise. From Ashworth & Chambers (2000), modified.

Fig. 7.

Symplectites of clinopyroxenes and magnetite in Lilloise olivine. Optical micrographs, plane polarized light. A section parallel to the platelet plane (100) of olivine. From Ashworth & Chambers (2000), modified.

Fig. 7.

Symplectites of clinopyroxenes and magnetite in Lilloise olivine. Optical micrographs, plane polarized light. A section parallel to the platelet plane (100) of olivine. From Ashworth & Chambers (2000), modified.

 
formula
The magnetite and clinopyroxene are formed from the reactants in solid solution in olivine. The previous reaction is not a redox one. The proportion of pyroxene over magnetite produced is 3 : 1 in terms of oxygen atoms and approximately in volume ratio. TEM observations indicate that the reaction is topotactic, both symplectite minerals having crystallographic orientations related to that of olivine and very a similar oxygen array (Moseley, 1984). A parental relation is found between close-packed planes in olivine and product minerals (Ashworth & Chambers, 2000). The reaction front as well as the grain boundaries are semicoherent. This implies that oxygen is an immobile component. So it is sensible to suppose that Ca, Fe, Mg can diffuse through the olivine and that Si is conserved in the reaction front. In the reaction front a flux of cations must originate between the crystallisation domains of clinopyroxene and magnetite.

Theory of intergrowth spacing

The model proposed by Ashworth & Chambers (2000) develops a work by Cahn (1959) from General Electric Research Laboratory, on the cellular segregation reactions.

Consider a region made of a sequence of parallel lamellar phases α, β produced by the transformation of a matrix phase m (of the reactant minerals) (Fig. 8). The composition of the transformed region is equal to that of the untransformed matrix. It is made by cells whose repeating period is the lamellar spacing. Within a cell all the crystals of one type are approximately parallel and have the same lattice orientation. Concepts such as nucleation and growth rate may be applied to the cell as a whole, as well as to individual crystals within the cell. As the cell grows, the individual plates maintain an approximately constant spacing so that branching and/or nucleation of new plates must occur. It is appropriate to distinguish sideways growth, which requires the production of new lamellae by nucleation and/or branching, and edgewise growth, which is the extention of the cell at the growth front towards the matrix (Christian, 1975). Ashworth & Chambers (2000) are concerned with edgewise growth.

Fig. 8.

Geometry of the growing symplectite: λ is the recurrence distance of the lamellae of α and β phases; δ is the thickness of the diffusion layer between lamellae and the reactant minerals of the matrix. Pα is the volume proportion of α. From Ashworth & Chambers (2000), modified.

Fig. 8.

Geometry of the growing symplectite: λ is the recurrence distance of the lamellae of α and β phases; δ is the thickness of the diffusion layer between lamellae and the reactant minerals of the matrix. Pα is the volume proportion of α. From Ashworth & Chambers (2000), modified.

However, cases are known where the rates of sideways and edgewise growth are similar and the lamellar spacing is a sensitive function of the temperature. The situation we are considering is the one in which the solute segregation depends only on diffusion within a layer, thickness δ, at the interface between cells and matrix (Ashworth & Chambers, 2000).

graphic
and
graphic
are the uniform concentrations of component i in phases α and β, respectively.  
formula
is the matrix composition up to the reaction front: transfer of matter from the matrix to the δ layer determines the growth rate V of the cells. If the diffusion in the matrix were sluggish,  
formula
should be calculated using the appropriate boundary conditions. Concentration gradients set in the δlayer, because the components' concentration in α,β are different. Diffusion flow transfers atoms parallel to the interface between cells and matrix. At a fixed temperature, pressure and mineral composition, the affinity of the overall reaction is −ΔG, a constant. It is made of several contributions. At first suppose a closed system whose boundary advances by δz. The Gibbs function increases by 
formula
(30)
where - ΔGrf is the affinity per mole of the reaction at the reaction front, Gdif is the affinity of the diffusion at the reaction front, lδz is the new interface area formed between α|βas a consequence of an advancement δz of the growth front, λ the cell spacing, v the mean molar volume of the simplectite and δ the interface energy. Then, ∆Grf > 0 if ∆Gdif does overcome a critical value 
formula
(31)
That is,  
formula
is the driving force  
formula
needed to form a unit area of a new interface at the grain boundary α|β. If the system is open the total affinity is: 
formula
(32)
The contribution -∆Gext is due to transfer of components from the exterior (the matrix) towards the reaction zone. It is assumed that the growth rate edgewise depends linearly on the flux of matter from the exterior towards the reaction front of thickness δ, and because the fluxes are proportional to the affinity driving the symplectic growth, a linear relation holds between the affinity and the reaction front advancement velocity V: 
formula
(33)
c is a constant.

To calculate ∆Gdif we must know the diffusive flux, Ji, of the i components, parallel to the growth front and driven by difference in compositions between α and β phases. The two phases have a recurrence distance in the direction of the flux Ji so the flux has a periodicity (Fig. 9). The cell is composed by a host mineral (the thicker one, say β) and a lamella (a) whose volume proportion is Pα. Assuming that elements do not diffuse within the cell but in the reaction front only, taking the origin at half lamella, the components rejected from α (to be deposited in the host mineral β) have a maximal concentration at nλ, n integer, and a minimal concentration at

Fig. 9.

Variation of flux (a) and concentration (b) with distance X in the reaction front. From Ashworth & Chambers (2000), modified.

Fig. 9.

Variation of flux (a) and concentration (b) with distance X in the reaction front. From Ashworth & Chambers (2000), modified.

 
formula
, at the middle of βAs a consequence the flux of the ithcomponent, Ji, has a maximum at  
formula
and a minimum at  
formula
. The Ji are linearised and represented by an appropriate sequence of broken lines: by consequence in a slab thickness δ parallel to the growth front, the current is 
formula
(34)
between 
formula
and 
formula
(35)
between 
formula
The variation of  
formula
in the interval  
formula
, is obtained by integrating Equation 34 with the boundary condition  
formula
 
formula
(36)
The variation of  
formula
between 
formula
is obtained by integrating Equation 35 with the boundary condition  
formula
 
formula
(37)
According to Fick's Law (neglecting cross coupling Dij coefficients): 
formula
(38)
Using Equation 38, Equation 36 is integrated in the interval 0 < x < 
formula
, with the boundary condition  
formula
obtaining: 
formula
(39)
Equation 37 is integrated in the interval 
formula
, with the boundary conditiongiven by (39): 
formula
, obtaining: 
formula
(40)
The last equation holds for “immobile” elements like Si and O. For elements diffusing from the matrix or along the interface (which is excluded here) the term containing Pα should be multiplied by a factor taking into account the material lost. Because Equation 40 will be used for closed reaction zone, we prefer to keep a ligther notation. We are describing the concentration and the fluxes as continuous functions of a space variable. In such a case the entropy production per unit volume and unit time is: 
formula
(41)
andSumming up the contribution to σi due to diffusion at the growth front between  
formula
and  
formula
:
 
formula
(43)
or, with the appropriate substitutions: 
formula
(44)
In a volume  
formula
meter the entropy produced by diffusion is 
formula
. In the time needed for the growth of a unit volume of symplectite 
formula
the entropy produced is 
formula
Multipling this quantity by the molar volume of symplectite v and by the temperature T we obtain the affinity driving the formation of one mole of symplectite by diffusion at the growth front. 
formula
(45)
After integration it we obtain: 
formula
(46)
Summing over the number of diffusing components and multiplying and dividing by LSi, for later convenience, one gets: 
formula
(47)
The total affinity is obtained by summing all contributions: 
formula
(48)
which can be cast in the form: 
formula
(49)
In Equation 49 the correspondence is the following: 
formula
(50)
The constants a, b are given by: 
formula
(51)
 
formula
(52)

Equation 49 contains an implicit relationship between λ and V. Equation 49 is third order in λ: it can be solved for fixed values of a, b, c, ΔG and values of V increasing from zero to an upper limit. Beyond this limit no real solution for λ can be found. If we assume that this limit represents the optimal value of the cell growth rate V* then the largest real root of the cubic equation is the optimal value of λ, for which -∆Gdif has the higher value. As V increases, & decreases because the faster transport of matter requires shorter diffusion distances; in turn shorter λ values imply the increase of −ΔGgb. Finally −ΔGext increases while −ΔGdif decreases with increasing V so that −ΔG is constant.

Alternatively, it can be assumed that the rate of energy dissipation by transport within the reaction layer and exchange with the matrix is maximal. In this case the value of V must be found which makes a maximum V(−ΔGdiff− ΔGext), whose expression is obtained using Equations 33, 47, 52: 

formula
(53)
Differentiating in respect to V the right-hand side member of Equation 53 and equating the result to zero, we obtain: 
formula
(54)
The expression of  
formula
is obtained by differentiating Equation 49 at constant ΔG: 
formula
(55)
Substituting Equation 55 in Equation 54 and multiplying by V the resulting expression, we obtain: 
formula
(56)
So the maximum occurs when 
formula
(57)
The relationship (57) between growth rate and spacing λ, holds when the rate of energy dissipation is maximal. It can be written as: 
formula
(58)
The condition of maximum in terms of AG is (Ashworth & Chambers, 2000): 
formula
(59)
which gives the relationship equivalent to Equation 58: 
formula
(60)
in both cases: ΔGext = 0 and ΔGext 0. It also indicates that half the affinity at the growth front drives diffusion and half is spent to increase the interface between host and lamella.Equation 57 is the key one: it was used by Ashworth & Chambers (2000) to characterise the symplectites in olivine. This would be an easy task when a, b are known. Looking at Equation 51 and 52 we see that some quantities can be measured, some others like the reaction layer thickness δ, are related to the approximated description of the mechanism of the symplectic reaction, while the interface energy γ and the phenomenological coefficients Lk must be known a priori. Unfortunately, this is not the case: researches are under way (an example will be seen in the next chapter) in order to know the relevant mineral and transport properties needed to exploit in full the constraints obtainable from rock texture and composition. To overcome these difficulties Ashworth & Chambers (2000) introduced some approximations and used independent estimates of the growth velocity V in order to obtain a lower bound to LSiδ. In particular, from geological information it is estimated V >6 × 10−16 m3s−1; from experiments on olivine oxidation γ< 0.3 Jm−2; it is measured λ≈ 4 μm. Finally, considering the term containing LSi as the only one contributing to b, the sum in (52) reduces to: 
formula
Identifying α with magnetite,  
formula
. From Equation 57: 
formula
being  
formula
as results from the measured Pα ≈ 0.25 and the concentration of Si in olivine. There are experimental indications that the calculated value of LSiδ occurs in the temperature range 800-1000 °C.

It is worth to report some Authors' considerations:

  • diffusion strictly parallel to the growth front can occur in fluid undersaturated systems;

  • high difference in composition between product minerals, particularly in the case of slow diffusion elements, favours symplectite formation.

Final considerations: the work is based on the assumption that the product of the symplectite growth rate by the diffusion affinity is maximum, but from the thermodynamics of irreversible processes a minimum would be expected at the steady state: so more fundamental arguments should be desirable also to explain why a periodic concentration over the growth front is stable.

An estimate of intergranular diffusion of Al in fluid undersaturated systems

Carlson (2002), using numerical simulations of coupled intergranular and intracrystalline diffusion processes in coronal textures around partially resorbed garnet crystals (Fig. 10), obtained a very precise estimate for the rate of intergranular diffusion of Al in the fluid-undersaturated system described in the following paragraph. The model is applied to partially resorbed garnets in mafic rocks of the Llano Uplift (Texas, USA).Actually these garnets are surrounded by layered coronal reaction textures and exhibit reversal intracrystalline diffusion profiles at their rims (Fig. 11). These textures are a consequence of a two-stage metamorphic event. In the first (prograde metamorphism), the garnets crystallised under conditions transitional among the amphibolite, granulite and eclogite facies, reaching peak temperature of about 750 °C, sufficient to nearly homogenise original growth zoning profiles in all but the largest garnets (Carlson & Schwarze, 1997). In the second stage (retrograde metamorphism), recrystallisation took place at lower pressures and under static conditions that allowed development of (i) coronal textures between garnet and sodic pyroxene, and (ii) reversal zoning in garnet. The coronitic structure, shown in Figure 10, from garnet to sodic pyroxene, is made up by: symplectite of plagioclase and amphibole; layer of amphibole and plagioclase; symplectite of plagioclase, amphibole and orthopyroxene; intergrowth of plagioclase and secondary sodic augite after original omphacitic clinopyroxene. Such features are the result of the coupling of dissolution reactions and intracrystalline diffusion in garnets during retrograde metamorphism. The steepness of the compositional profiles near the garnet rim (Fig. 11), depends upon the relative rates of the dissolution reaction and the intracrystalline diffusion that results from it. The relative rates of these processes are assessed by a numerical simulation of multicomponent intracrystalline diffusion in garnet, based on the computation of garnet size variation and compositional changes in Fe, Mg, Mn, Ca in response to the variations of temperature during retrograde metamorphism. The Carlson (2002) simulation is based on the following assumptions.

Fig. 10.

Photomicrograph with polarised light (uncrossed polars) of layered coronal reaction zone. Relict garnet is at bottom of photo. Next layer upward is a symplectite of plagioclase and amphibole; its upper limit, marked by a solid white line, is the original boundary between the reactants garnet and omphacite. Above there is a layer of amphibole, with a narrow band of plagioclase near its centre; this borders on a brownish zone that comprises a symplectite of plagioclase, amphibole and orthopyroxene. The dashed line marks the boundary with a top zone of intergrown plagioclase and secondary sodic augite after original omphacitic clinopyroxene. From Carlson (2002), modified.

Fig. 10.

Photomicrograph with polarised light (uncrossed polars) of layered coronal reaction zone. Relict garnet is at bottom of photo. Next layer upward is a symplectite of plagioclase and amphibole; its upper limit, marked by a solid white line, is the original boundary between the reactants garnet and omphacite. Above there is a layer of amphibole, with a narrow band of plagioclase near its centre; this borders on a brownish zone that comprises a symplectite of plagioclase, amphibole and orthopyroxene. The dashed line marks the boundary with a top zone of intergrown plagioclase and secondary sodic augite after original omphacitic clinopyroxene. From Carlson (2002), modified.

Fig. 11.

Representative fits of modelled diffusion profiles to measured compositions in relict garnet. Circles are microprobe analyses; dotted lines: extrapolations used to estimate original profiles; solid lines: the results of the numerical simulation. From Carlson (2002), modified.

Fig. 11.

Representative fits of modelled diffusion profiles to measured compositions in relict garnet. Circles are microprobe analyses; dotted lines: extrapolations used to estimate original profiles; solid lines: the results of the numerical simulation. From Carlson (2002), modified.

  1. 1.
    The temperature of the rock is decreasing, at constant pressure of 3 kbar, according to the linear equation: 
    formula
    (61)
    where T0 is the initial temperature, namely the temperature at which coronitic reaction begins and the rate is expressed in  
    formula
    .
  2. 2.

    The garnet is a sphere whose initial radius (before dissolution) is considered to be the upper limit of plagioclase and amphibole symplectite (Carlson & Johnson, 1991). It is shown in Figure 10 by the solid white line.

  3. 3.

    The initial element concentration profile of the garnet is obtained by extrapolating the supposed unaffected composition of the interior portion of the crystal to the portion of the garnet dissolved (dotted line in Fig. 11).

  4. 4.
    The garnet dissolution rate is limited by diffusion of Al across the product zone. Therefore, diffusional control of the reaction kinetics justifies the assumption of the Arrhenius law to determine the rate of change of garnet volume  
    formula
    :
 
formula
(62)
where QIGD is the activation energy for intergranular diffusion of Al: from a previous work its value results 
formula
. Because the integral of Equation 62 over the dissolution interval must equal the total volume of garnet resorbed, the value of  
formula
is determined by the volume loss of the crystal ΔV: 
formula
(63)
where t25 is the time at which T = 25 °C, i.e. the time at which dissolution is considered to be complete.

The simulation is iterative. Given an initial temperature T0 and the thermal history described by Equation 61, the garnet (with radius and composition determined as explained in points 2 and 3) is resorbed by the amount required by Equation 62, and the time for each volume decrement is computed from (61). For each increment of dissolution, the flux of material into or out of the rim of the remaining crystal is determined by a retention/loss factor. The lattice diffusion in garnet is computed solving the multicomponent diffusion equation, adapting values of the diffusion matrix from Chakraborty & Ganguly (1992) and accounting for their composition dependence following Lasaga (1979). The dissolution/diffusion cycle is repeated until the radius of the model crystal matches that of the relict crystal. The time tf resulting from this procedure gives the time required to resorb 95% of the measured volume lost by garnet.

Iterative calculations are required to determine the retention/loss factors. To begin, an initial trial value for each element (Fe, Mg, Mn, Ca) is arbitrarily chosen. For instance, if at the end of the simulation the crystal contains an excess of one element with respect to the total observed amount, then for the next simulation a smaller retention factor is considered.1

Now, it is possible to calculate the effective intergranular diffusivity (strictly not the grain boundary diffusion coefficient because an intergranular medium in which Al could be soluble was possibly present) of Al (DAlef) by equating the measured length scale for intergranular diffusion in the coronal textures with the characteristic diffusion distance, travelled by diffusing atoms: 

formula
(64)
The value of x is taken as the distance between the surface of the reactants (between the edge of residual garnet and the dashed line in Fig. 10) in the corona when the reaction ceased. Fourteen compositional profiles were modelled by Carlson (2002), selected from specimens that represent a wide range of possible dissolution histories, in order to asses the uncertainties that arise from natural variations. But all fourteen modelled profiles yielded very similar results, indicating that the approach is precise. Fitting was done by adjusting only the model parameters T0 (655-686 °C); all other parameters are identical in all 14 fits. The estimated intergranular diffusivity for Al, resulting from this procedure, is: 
formula
(65)
This equation is only applicable to hydrous but fluid undersaturated systems and in the range of temperature encompassed in these models (500-650 °C).

The ultrahigh pressure coronitic reactions in a metagranodiorite

Geology and petrography

Metagranodiorite samples from the Brossasco-Isasca Unit (Biino & Compagnoni, 1992; Bruno et al., 2001), Dora-Maira Massif, western Alps, show pseudomorphous and coronitic textures where igneous minerals were partially replaced by ultrahigh pressure (UHP) metamorphic assemblages. The Brossasco-Isasca Unit (BIU) is a slice of Variscan continental crust recrystallised under UHP metamorphic conditions during the Alpine orogeny (Henry, 1990; Chopin et al., 1991; Compagnoni et al., 1994, 1995; Compagnoni & Rolfo, 2003). The metamorphic peak is estimated by many workers at P = 33±3 kbar and T = 750 ± 30 °C (Chopin, 1984, 1987; Chopin et al., 1991; Kienast et al., 1991; Sharp et al., 1993; Compagnoni et al., 1994), and, recently, by Hermann (2003) at about 43 kbar and 750 °C. The metagranodiorite originally consisted of quartz, plagioclase, K-feldspar, biotite and accessory apatite, zircon and a Ti-rich phase, most likely ilmenite. During the Alpine polyphase metamorphism, the igneous minerals were (i) replaced by polycrystalline aggregates of metamorphic minerals, (Site P; Fig. 12), (ii) reequilibrated to metamorphic compositions, e.g. biotite, or (iii) reacted and developed coronitic structures between biotite and adjacent minerals, e.g. between biotite and quartz (Site A, Fig. 13), biotite and K-feldspar (Site B, Fig. 14), and biotite and plagioclase (Site C, Fig. 15). Representative analyses of the phases are reported in Bruno et al. (2001). The plagioclase (Site P) is replaced by a polycrystalline aggregate of zoisite + jadeite + quartz/coesite + kyanite + K-feldspar to a pseudomorph (Fig. 12). Zoisite and kyanite are pure, and K-feldspar is low in albite (Or90Ab10). Na-pyroxene is a solid solution of jadeite, Ca-Tschermak and Ca-Eskola (Bruno et al., 2002), but usually it is partially replaced by retrograde albite or oligoclase. At the original igneous biotite-quartz contact (Site A), a single continuous corona of weakly zoned garnet, with composition Alm76-78Prp21-23Grs1-2Sps1-2, develops (Fig. 13). Usually, garnet is rimmed by a retrograde biotite. Between biotite (partially replaced by phengite I) and K-feldspar (Site B), the following composite corona is formed (Fig. 14):

Fig. 12.

Plagioclase domain (site P). Aggregate of Ab + Jd + Zo + Ky + Kfs + Qtz, pseudomorphically replacing the original igneous plagioclase.

Fig. 12.

Plagioclase domain (site P). Aggregate of Ab + Jd + Zo + Ky + Kfs + Qtz, pseudomorphically replacing the original igneous plagioclase.

Fig. 13.

Site A: Biotite partly replaced by phengite. Note the development of a continuous garnet corona against quartz.

Fig. 13.

Site A: Biotite partly replaced by phengite. Note the development of a continuous garnet corona against quartz.

Fig. 2.

Site B: Relict of igneous biotite included in K-feldspar. Biotite is partly replaced by phengite (PhI). Note the development of a composite corona of garnet, garnet+quartz and quartz+phengite (PhII).

Fig. 2.

Site B: Relict of igneous biotite included in K-feldspar. Biotite is partly replaced by phengite (PhI). Note the development of a composite corona of garnet, garnet+quartz and quartz+phengite (PhII).

  • a continuous corona of weakly zoned garnet (Alm78-80Prp15-17Grs3Sps2);

  • a continuous corona of garnet (Alm78–80Prp17-19Grs2-3Sps0-1) with vermicular quartz inclusions, elongated perpendicular to the corona;

  • a continuous corona of a quartz-phengite (PhII) symplectite in continuity with the garnet-quartz symplectite.

Between biotite and plagioclase (Site C), the following mineral sequence is observed (Fig. 15):
Fig. 15.

Site C: Composite corona developed at the original contact between igneous biotite and plagioclase. The corona consists of garnet + quartz, phengite (PhII), and jadeite + garnet. Biotite is completely replaced by phengite (PhII).

Fig. 15.

Site C: Composite corona developed at the original contact between igneous biotite and plagioclase. The corona consists of garnet + quartz, phengite (PhII), and jadeite + garnet. Biotite is completely replaced by phengite (PhII).

  • a corona of garnet (Grs49-50Alm42Prp7-8Sps0-1) plus quartz;

  • a composite corona of idioblastic garnet (Grs62–84Alm16-32Prp0-6) and jadeite. From biotite towards plagioclase, the relative amount of garnet decreases and that of clinopyroxene increases. Garnet becomes richer in grossular and pyroxene richer in jadeite. Coronitic garnet is always asymmetrically zoned with Ca increasing and (Fe + Mg) decreasing, from biotite towards plagioclase. A retrograde phengite II developed outside the garnet corona.

Equilibrium thermodynamic modelling

Information on the metamorphic history of the metagranodiorite may be obtained by integrating petrographic observations of multivariant mineral associations with calculations of relative stability and composition of phases. To perform equilibrium calculations it isessential to determine the volume of the rock which reached chemical equilibrium. The bulk composition of this volume defines the effective bulk composition (EBC). In the case of the Brossasco-Isasca metagranodiorite it is impossible to define only one EBC, because the absence of a pervasive deformation during metamorphism prevented the exchange of matter among the different portions of the rock. This means that the original igneous mineral modes and mineral chemistry determined sites with different EBC (sites A, B and C), which evolved independently during metamorphism, at least as a first approximation. In order to establish the extent of equilibrium or disequilibrium and to understand the mineralogical evolution of each site during prograde metamorphism, the composition and abundance of local phases have been used as a basis for thermodynamic calculations.2 The initial compositions for the three considered sites are given in Table 1.

Table 1.

Initial phase compositions for the three sites considered. The moles of phase components are normalised to 20 moles of cations.

PhasePhase componentsSite ASite BSite C
BtAnn0.2200.2200.088
Phl0.1500.1500.060
Eas0.1200.1200.048
Mn-bt0.0100.0100.004
Qtz14.00
KfsAb0.0300.600
Or0.1602.390
An0.0100.010
PlAb2.448
Or0.072
An1.080
PhasePhase componentsSite ASite BSite C
BtAnn0.2200.2200.088
Phl0.1500.1500.060
Eas0.1200.1200.048
Mn-bt0.0100.0100.004
Qtz14.00
KfsAb0.0300.600
Or0.1602.390
An0.0100.010
PlAb2.448
Or0.072
An1.080

The equilibrium calculations have been performed by minimising the Lagrangian, 

formula
(66)
at constant temperature and pressure (Smith & Missen, 1982). G is the Gibbs function: 
formula
(67)
A (l, m) is the formula matrix, l is the number of elements, m is the number of phase components, b(l) is the element abundance vector, λ(l) is the row vector of Lagrange multipliers, μi is the chemical potential of each end-member component in each phase, and ni are their mole numbers which are constrained by the mass balance equation (An - b = 0) and non-negativity conditions (ni > 0); ñ is the transpose of n.

The abundance and composition of the phases have been calculated along a P-T trajectory described by the following equation: 

formula
(68)
which corresponds to the prograde (subduction) trajectory suggested by Schertl et al. (1991) and Compagnoni et al. (1994) for the BIU. Two hundred points were calculated for the P-T region between 3 and 40 kbar, and 420 and 830 °C.

Thermodynamic modelling was undertaken in the KNCFMnMASH system with biotite (Bt), plagioclase (Pl), K-feldspar (Kfs), phengite (Ph), garnet (Grt), jadeite (Jd), kyanite (Ky), zoisite (Zo), quartz (Qtz), coesite (Coe) and water as phases. The phase components considered are: annite (Ann), phlogopite (Phl), eastonite (Eas) and Mn-biotite (Mn-bt) for biotite; albite (Ab), anorthite (An) and orthoclase (Or) for feldspar; muscovite (Ms), Fe-celadonite (Fe-cel) and Mg-celadonite (Mg-cel) for potassic white mica; almandine (Alm), grossular (Grs), pyrope (Prp), and spessartine (Sps) for garnet. The fluid is supposed to be pure H2O.

The thermodynamic properties were taken from the Holland & Powell (1990) database, updated by Vance & Holland (1993). The solid solution models for biotite and phengite are from Powell (1978), Holland & Powell (1990) and Vance & Holland (1993); for garnet from Berman (1990); and for feldspar from Fuhrman & Lindsley (1988).

By the equilibrium calculations the following information have been obtained.

  • In each site, different reactions were overstepped at different temperatures and pressures, and produced typical mineral assemblages. In sites A and B the partial replacement of the igneous biotite by phengite (Ph I) and the development of a garnet corona suggest the following reaction: 
    formula
    (69)
    where 
    formula
    (70)
    with n0Bt being the initial amount in moles of igneous biotite, X0Eas the initial molar fraction of eastonite in igneous biotite, XEas the final molar fraction of eastonite in metamorphic biotite and XMs the final molar fraction of muscovite. Reaction (69) stops when eastonite is consumed (XEas = 0). Instead, in site C the reactions producing garnet are (69) and: 
    formula
    (71)
     
    formula
    (72)
  • On a thermodynamic basis it has been assessed that, for peculiar bulk chemical compositions, biotite is stable all along the calculated P-T trajectory (Fig. 16). This is evident by considering the reactions producing garnet in the different sites. In sites A and B, the eastonite content determines the biotite abundance. Indeed, reaction (69) is the only net-transfer reaction involving biotite. While in the site C there are two net transfer reactions (69) and (71) able to consume completely the biotite. Likewise, both microscopic observations and thermodynamic calculations indicate that only water-constant reactions occurred, such as observed in the quartz eclogite facies metagranodiorite from Monte Mucrone, Sesia zone (Rubbo et al.,1999).

  • Thermobarometric estimates, obtained by comparing calculated and measured garnet compositions, indicate that the Brossasco metagranodiorite retains evidence of recrystallisation at a minimum pressure of 24 kbar at 650 °C. As an example, the calculated garnet composition for site A is reported in Figure 17: the measured composition (Alm76-78Prp21-23Grs1-2Sps1-2) matches the calculated at 24 kbar. Thermodynamic calculations also show that the Brossasco metagranodiorite experienced no further reactions for the P-T range between 28 kbar and 40 kbar and 690 to 830 °C (Fig. 16).

Fig. 16.

Site A: Variations of abundance of biotite, phengite, garnet and quartz-coesite as a function of pressure and temperature.

Fig. 16.

Site A: Variations of abundance of biotite, phengite, garnet and quartz-coesite as a function of pressure and temperature.

Fig. 17.

Site A: Variation of garnet composition as a function of pressure and temperature.

Fig. 17.

Site A: Variation of garnet composition as a function of pressure and temperature.

Therefore it can be concluded that coronitic textures from eclogite facies metagranitoids preserved within greenschist to amphibolite facies orthogneiss can provide petrologic information useful to reveal the former presence of HP or UHP metamorphic recrystallisation.

Further considerations on the model and the metamorphic evolution of the eclogite facies metagranodiorite

Since the equilibrium calculations have only been performed along the prograde P-T path (Eqn. 68), only qualitative discussions on the retrograde metamorphism suffered by the metagranodiorite can be done. The major disequilibrium feature acquired during retrogression is revealed by the discrepancy between the calculated and measured compositions of biotite and phengite. The measured biotite compositions are inconsistent with that of garnet still retaining the composition of high pressure. Such discrepancies can be explained by considering that (i) the retrograde trajectory is different from the prograde one, and (ii) the sluggish volume diffusion limits the rate of the rock re-equilibration. During cooling, at subsequent steps successively large parts of garnet grains are effectively isolated from the reacting rock volume and the EBC will change. Such process is strongly stressed by the quick exhumation rate of the rock, estimated at about 2 cm/year by Gebauer et al. (1997) and 3 cm/year by Duchêne et al. (1997). (See Rubbatto et al., 2003, in this volume for further details.) As demonstrated by Bruno & Rubbo (unpublished results), the rapid cooling implies a limited volume diffusion in the coronitic garnet of the metagranodiorite, therefore, only a thin rim of garnet was in equilibrium with the adjacent phases (biotite and/or phengite) during retrogression. Therefore, re-equilibration of biotite and phengite occurred along a trajectory and with an EBC different from those of their formation. Moreover, there are some lines of evidence that metamorphism was not strictly isochemical: some matter, exchanged among sites, modified the initial EBC. Indeed, site A is characterised by Ca-free phases but garnet contains Ca; zoisite grew at the expense of the original plagioclase (site P) but hydroxyl groups diffused from close-by sites; the polyphase reaction corona at site B cannot be balanced using a simple reaction among the phases observed, but a flux of matter from outside the system is required. For this symplectite reaction the model of Ashworth & Chambers (2000) previously described can explain the formation of this corona, where a periodicity in the lamellar minerals is observed.

In order to avoid misleading interpretations, all these factors should be considered when the equilibrium calculations are used to derive petrological information from the mineral assemblages. It is also worth remembering that the model system here presented lacks accurate thermodynamic knowledge on some phases especially on aqueous fluid rich in Na, K, Si and Al. As shown the Hermann's (2002) experiments, these cations greatly affect the behaviour of fluids at high P and T in alkali-rich systems, such as the metagranodiorite studied.

Garnet growth model

Zoned coronitic garnets (Figs. 18, 19) developed between igneous biotite and plagioclase in the Monte Mucrone metagranodiorite (MMM) from the Sesia-Lanzo Zone, Western Alps. The MMM crops out in the “Eclogitic Micaschist Complex” (EMC), which is one of the main subunits making up the Sesia Zone (Dal Piaz et al., 1972; Compagnoni & Maffeo, 1973; Compagnoni, 1977; Compagnoni et al., 1977). The Sesia Zone is characterised by the widespread occurrence of eclogite facies assemblages in a wide spectrum of continental crustal lithologies. The EMC of the Sesia Zone is a fragment of Variscan continental crust, which was metamorphosed during the early Alpine HP subduction event (Oberhänsli et al., 1985). The early Alpine HP stage is constrained at T = 500-600 °C and P = 16-20 kbar (Compagnoni, 1977; Lardeaux et al., 1982; Droop et al., 1990).

Fig. 18.

SEM backscattered image of the Monte Mucrone metagranodiorite. Biotite is partially replaced by phengite, and a garnet corona developed between biotite and plagioclase, now replaced by a polycrystalline aggregate of Zo + Jd(Ab) + Qtz.

Fig. 18.

SEM backscattered image of the Monte Mucrone metagranodiorite. Biotite is partially replaced by phengite, and a garnet corona developed between biotite and plagioclase, now replaced by a polycrystalline aggregate of Zo + Jd(Ab) + Qtz.

Fig. 19.

Measured compositional pattern. Spot analyses are 2 μm spaced. Bt, Core and Pl correspond to the garnet-biotite interface, the garnet core and the garnet-plagioclase interface, respectively. From Rubbo et al. (1999), modified.

Fig. 19.

Measured compositional pattern. Spot analyses are 2 μm spaced. Bt, Core and Pl correspond to the garnet-biotite interface, the garnet core and the garnet-plagioclase interface, respectively. From Rubbo et al. (1999), modified.

The garnet corona (Fig. 18) is made up by aggregates of zoned garnets. Afingerprint of the metamorphic evolution is the Ca concentration in garnet, which is very low in the core and sharply increases towards the rim. A magnification of the X-ray compositional maps shows islands of nearly concentric iso-concentration lines (Rubbo et al., 1999), suggesting a nucleation and growth of single crystal, which later coalesced to a continuous layer. The equilibrium thermodynamic calculations show that garnet is stable over a wide range of pressure and temperature, as in the Brossasco-Isasca metagranodiorite previously described: so the isothermal isobaric model of layer formation is not applicable. Moreover, the pattern of the grossular concentration is an indication that the rate of Ca release and breakdown of plagioclase determined an increase in the garnet growth rate and Ca incorporation.3 It is also expected, as confirmed by an estimate (Rubbo et al., 1999), that the diffusion in the growth medium was faster than in garnet. Finally, it is sensible to suppose that the composition of the outer layer of garnet was close to, but not in equilibrium with, the other minerals, while temperature and pressure were changing during the metamorphism. An adjustment of the surface layer to the prevailing conditions could be achieved by an exchange of matter between garnet and the growth medium not involving its growth followed by segregation of the elements uptaken. Exchange reaction are also implied when geothermometers are used.

From these considerations, it follows that thermobarometric information can be retrieved from the zoning pattern if some kinetic hypotheses can be made on the rate limiting processes. The qualitatively known P-T trajectory can be used for a consistency test.

The calculation technique, used to simulate the garnet growth, is based on a work by Small & Ghez (1980), where it is shown how the equilibrium thermodynamics can be used to approximate the solution of a diffusion and growth problem. We expose this approach describing the growth of a spherical garnet (solid solution of pyrope, almandine, grossular and spessartine) and the evolution of the abundance and composition of minerals associated with it along a P-T trajectory. The calculations start at given initial amounts of phases (rock volume from which the garnet grows), determining the initial EBC, and at given initial values of temperature (T0) and pressure (P0). The initial EBC is made by the igneous phases with a composition typical of a granodiorite while the abundance is estimated from the phase abundance in the site of the garnet (Table 2). However, plagioclase is not included in the initial EBC.

Table 2.

Initial moles of end members in the EBC.

AnnPhlEasMn-btAbOrAnQtz
0.18620.141140.172.66−30.150.600.00.5
AnnPhlEasMn-btAbOrAnQtz
0.18620.141140.172.66−30.150.600.00.5

Temperature and pressure are iteratively changed by small increments, δT and δP, along an elliptic P-T trajectory. Indeed, being interested in both prograde and retrograde paths the non linear trajectory proposed by Compagnoni et al. (1995) has been parametrised as an elliptical arc. The trajectory parameters are allowed to vary within the constraints imposed by the Compagnoni's et al. (1995) work. 

formula
(73)
 
formula
(74)
 
formula
(75)
 
formula
(76)
 
formula
(77)
Along with temperature and pressure the initial EBC progressively changes because plagioclase is progressively added and very small amounts of garnet, after its nucleation, are subtracted from the EBC, according to the following equation: 
formula
(78)
nk stands for the vector whose elements are (nθi)k, where θ runs on the phases, i runs over the phase components, and k is associated to the iteration4. Mϕ(i.e. Mg for the garnet and Mp for plagioclase) is the integral of the growth/dissolution rate of the crystal, at temperature Tk and pressure Pk over a time Δt. If is positive, the phase ϕ grows, whereas it resorbs if Mϕ is negative.

To generate a spherical crystal of garnet, the moles Mg are homogeneously distributed as concentric shells. The total volume of segregated garnet is divided by a suitable number of crystals, to reproduce the size of the measured growth sector.

Mϕ is calculated as a balance between the number of moles added, Eϕ(Tk, Pk, nk), and subtracted, (Tk− δT, Pk- δP), to phase ϕ. Eϕ(i.e. Eg for the garnet) is the number of moles of the outer shell of phase ϕ having the equilibrium composition at temperature Tk, pressure Pk, when the moles of end-members making the EBC are nk.

Eϕ is calculated at every iteration by minimising the Gibbs function of the system. The details of the equilibrium calculations and related references, are reported in Rubbo et al. (1999).

Nϕ is a function of Arrhenius type. It is calculated at (Tk - 5T, Pk − δP) and it is the number of moles of phase ϕ to be resorbed at Tk, Pk.

The explicit expressions of Nϕused are: 

formula
(79)
 
formula
(80)
T0 = 608 K is the lowest temperature of the considered prograde trajectory and Tk the temperature of the kth iteration. To increase the input of Ca in the EBC, due to the dissolution of plagioclase, Np is made increasing with temperature. Conversely, Ng is made as a decreasing function of temperature, favouring garnet growth with increasing temperature. The small amount of garnet resorbed (and plagioclase redeposited) allows for a re-equilibration of garnet (plagioclase) involving not only its surface. This also comply with the Small & Ghez (1980) calculation strategy, to represent diffusion limited growth (dissolution).

There are no constraints on the values of the parameters αp, αg, βp, βg: so, they are determined by trial and error until the calculated concentrations satisfactorily fit the measured profiles. The optimal values are: αg = 0.12, βg = -14, αp = 2.5, βp = -5000.

The Monte Mucrone metagranodiorite (MMM) garnet zoning reproduced with this model is reported in Figure 20. The calculated garnet zoning best compares with the measured one and is a confirmation that the kinetic assumptions made are correct. This garnet growth simulation has allowed to obtain the following information:

Fig. 20.

Calculated garnet zoning. It must be compared with the measured zoning (Fig. 19) of the sector from the garnet core to the face towards plagioclase. From Rubbo et al. (1999), modified.

Fig. 20.

Calculated garnet zoning. It must be compared with the measured zoning (Fig. 19) of the sector from the garnet core to the face towards plagioclase. From Rubbo et al. (1999), modified.

  • The MMM garnet records a typical clockwise P-T path for regional metamorphism, where pressure peak (P = 17 kbar, T = 835 K) precedes thermal peak (P = 16 kbar, T = 862 K).

  • The maximum growth of garnet occurs close to the pressure peak and then the garnet resorbs during the retrograde path, but dissolution is interrupted by growth episodes.

  • It has not been possible to obtain the decrease of pyrope component towards the garnet rim by changing the parameters. This discrepancy between calculated and measured zoning may be caused by the coalescence of garnets forming the corona, which hindered the grain boundary diffusion of Mg from the garnet/biotite to the garnet/plagioclase interface. Because in the simulation the grain coalescence is not considered, the calculated molar fraction of Mg is somewhat higher than that measured in the outermost garnet layers towards the biotite/plagioclase interface. Then, the zoning simulation provides, indirectly, evidence of a factor of disequilibrium during garnet growth. Evidences of element disequilibrium in rocks have been reported in many works (e.g., Carlson, 2002; Hirsch et al., 2003). It appears that partial disequilibrium (meaning disequilibrium for some elements, but not for others) may be a common phenomenon during metamorphic mineral growth, even in ordinary prograde reactions that progress to completion. In the MMM, Mg seems to be unable to equilibrate at the scale of the coronitic structure, as well as Ca, indicating the following relation between the interdiffusion coefficients in the grain boundaries: DCa < DMg < DFe = DMn.In a recent research (Bruno & Rubbo, unpublished results) it was found that the agreement between calculated and measured pyrope profile is improved when lattice diffusion in garnet is considered.Some decrease of the pyrope component concentration at the garnet rim, could possibly be obtained by changing the retrograde P-T trajectory, but this analysis has not been undertaken.

  • Lattice diffusion in garnet occurred to a limited extent and was accompanied by important fluctuations of composition at the garnet growth front. The component concentration profile changes sharply near the garnet rim: this sharp change occurred at the high temperature segment of the P-T path and cation diffusion could then be active. For this reasons, a simulation of the growth accompanied by lattice diffusion in garnet has been undertaken: the correlation between temperature, pressure and time has been obtained and the garnet growth rate has been estimated by Bruno (2002).

Appendix

Some notions of irreversible thermodynamics

Let us consider a closed system in absence of external fields, capable of p-V work only. The first principle postulates the existence of a function of the state of the system, the internal energy (E), whose variations are due to the heat dQ and mechanical work -pdV exchanged with the surroundings through the system boundaries. dQ is positive if heat is received by the system. 

formula
(1)
The internal energy can change by interactions between the system and the surroundings but not by transformations within the system. The internal energy of an isolated system cannot change. 
formula
(2)
The indexes e, i, here and r in the following, will be used to indicate exchanges with the surroundings (e), exchanges within the system between phases (i), and mole variation as consequence of chemical reactions (r). For open regions can be defined as the work done on the region if it were closed (Haase, 1969), that is, in the simple situations here considered, the mechanical work dw = -pdV.

Let consider a system made of two phases and let assume that no chemical reactions occur. External fields are absent. If the system is open, the “heat” adsorbed during an infinitesimal state change, by one of its open c-component phase is: 

formula
(3)
hk is the partial molar enthalpy of species k and denk, dtnk are the amount of species k exchanged with the surroundings and with the other phases, respectively.

If the system is closed and the two homogeneous phases a, E are at uniform temperature and pressure Ta, pa and TE, pE, respectively, matter can be exchanged between phases only (denk = 0). 

formula
(4)
From Equation 3 written for every phase we obtain: 
formula
(5)
 
formula
(6)
The heat can be partitioned: that exchanged with the surroundings and that within the system between the two phases: 
formula
(7)
 
formula
(8)
The heat exchanged between the whole system and the surroundings is deQα + deQβ. The variation of the whole system's internal energy is: 
formula
(9)
The energy conservation requires: 
formula
(10)
and thus substituting (5) and (6) in (7) and (8) and the result in (10) we find: 
formula
(11)
This relation says that the energy received by phase α from phase β is of equal amount but of sign opposite to the energy recieved by β from α. The heat transfer between the two phases is, by using Equations 4 and 11: 
formula
(12)
Only if  
formula
(Haase, 1969). In the case the phases cannot exchange matter between them, it is: 
formula

The second law of thermodynamics states that the change of entropy of a system is a balance between two terms: the entropy flux due to exchanges with the surroundings (deS) and a non-negative source term due to irreversible processes occurring within the system (diS), that is 

formula
(13)
 
formula
(14)
If the system experiences only reversible modifications, then diS = 0.

Let's consider an isolated system composed of two phases αand β. For the whole system the following relation holds: 

formula
(15)
Within each phase, the production of entropy is non-negative: 
formula
(16)
A situation such that dS > 0 with 
formula
is impossible (Prigogine, 1968).

As an example, the entropy produced by the flux of heat caused by a difference of temperature is calculated for a system made by two closed phases a, E, kept at uniform temperatures Tα and Tβ. The system entropy is the sum 

formula
(17)
Every phase exchanges heat with the surroundings and with the other phase: 
formula
(18)
As a consequence of the first principle, we have: 
formula
and then: 
formula
(19)
These contributions can be grouped in the following way: 
formula
(20)
According to experience, if Tα< Tβthen di Qα>0 and entropy is produced by the internal process. When the temperature is uniform then diS = 0. The rate of production of entropy is: 
formula
(21)
Let us observe that  
formula
is the product of the rate (or flux)  
formula
with the function of state  
formula
, which is the “force driving the heat “flux .

As a further example, let us consider two subsystems or phases αand E making a system closed to matter exchange with the surroundings. Each subsystem has homogeneous temperature Tα, Tβ, pressure pα, pβ, and chemical potential  

formula
of the k component. Let suppose that between the two phases an exchange of matter occurs. For the c-component subsystem (dropping the superscipts for the moment) the first principle reads: 
formula
(22)
The Gibbs equation (obtained combining the first and second principles) is: 
formula
(23)
Now dnk ≡ dink describes the moles variation for the exchange within the system, between the phases. The entropy balance dS = dSα + dSβ (Eqn. 13), is calculated by Equations 23 and 22: 
formula
(24)
The rate of entropy production is calculated by introducing Equation 11: 
formula
(25)
Only if  
formula
and  
formula
is zero.

A further example is the entropy balance for a closed phase where a chemical reaction occurs. Using the reaction advancements ξ, it results  

formula
are the stoichiometric coefficients of the reacting species. Introducing the affinity (A) and the reaction rate (v) 
formula
the entropy balance reads 
formula
(26)
Being d S > 0, it ensues that αand v have the same sign; so if α< 0, then v < 0 and the reaction proceeds spontaneously towards the left (reactants). At equilibrium α= 0, v =0.

In the example seen and in general, each term in  

formula
is a product of a rate or generalised flux (Jk) and a state function which is a generalised affinity or driving force(Xk). In the example on the chemical reaction JK = v and  
formula
.

The rate of entropy production  

formula
has therefore the following structure: 
formula
(27)
At equilibrium both fluxes and driving forces are zero for every irreversible process: 
formula
(28)
A central point, suggested by experience, is the assumption that close to equilibrium the fluxes depend linearly on forces.

If one considers a system where two independent irreversible fluxes are produced by two independent forces, the hypothesis is made that 

formula
(29)
 
formula
(30)
 
formula
(31)
This assumption implies that if one driving force, say X1, is zero, the flow J1 0, being driven by X2. The two processes are coupled. Coupling is possible between processes occurring in the same part of the system and if the forces driving them have the same tensorial character (Curie's principle).5 For instance the affinity driving a chemical reaction (a scalar cause) may not be coupled with a temperature gradient, producing a heat flux. Substituting these expressions in Equation 27 one obtains: 
formula
(32)
 
formula
(33)
With reference to Equation 32 it can be shown that,  
formula
being positive for all positive and negative values of X1, X2, except X1 = X2 = 0, in which case  
formula
= 0, the following conditions hold: 
formula
(34)
To generalise using the mathematical language,  
formula
is a definite positive quadratic form.

A second central point is the theorem by Onsager, stating that the coupling coefficients are equal: 

formula
(35)
The Onsager relations are valid for the coefficients of the phenomenological equations if independent irreversible fluxes are written as linear functions of independent thermodynamic forces (see an example in Nishiyama, 1998). The calculation of the entropy production affords a means of obtaining the proper conjugate irreversible fluxes and thermodynamic forces needed to set up the phenomenological relations. We take relationship (35) as a postulate.

The previous equations give deep insight in the nature of steady states. In a steady state the state functions are independent on time but fluxes can go on within the system.

Particularly interesting are those states occurring when k independent affinities out of the n are constant. This will be shown with the help of Equation 33 in the case of two independent forces. In this particular case, a theorem, by Prigogine, can be stated as: if a system with two independent forces X1, X2, is kept in a state with fixed say X1, and minimum entropy production, the flux J2, conjugate to the non-fixed force, vanishes. Deriving Equation 32 with respect to X2, keeping constant X1, using Equation 35, one obtains: 

formula
(36)
Conversely if X1 is fixed and J2 = 0 the entropy production per unit time is minimal. Being  
formula
> 0, the condition (36) identifies a minimum.

To obtain (36) the Lij are assumed constant.

Up to now, examples of irreversible processes in so-called discontinuous systems have been shown. These systems are made of uniform phases but differences of some intensive properties can occur at the boundaries between phases. In continuous systems the physical properties depend on time and are continuous functions of the space coordinates. The extension of the previous results to continuous systems does not require new physical principles. However, the description of these systems demands a heavy mathematical formulation. We will not derive the equations for the continuous systems, but simply introduce the particular form of these equations as needed for discussing ordinary diffusion.

Diffusion

Diffusion is a way of transport of matter caused by concentration gradients, occurring in a wide variety of processes in earth sciences and in materials sciences. Multicomponent diffusion is the transport process occurring when the flux of a component is affected by the concentration gradients of a second component (Cussler, 1976, 1984). Examples of applications to earth sciences are: diffusion of components in the melts feeding crystals growing in magma chambers; diffusion in solution, along mineral interfaces and within minerals promoting reactions in rocks and changes in bulk composition by metasomatic and metamorphic processes (Lasaga, 1997).

A short introduction to a phenomenological description of diffusion will be presented here. Several textbooks and research papers are indicated to the interested reader: Kirkaldy & Young (1989) treats diffusion comprehensively, Crank (1999) is a good mathematical introduction; the important book edited by Ganguly (1991) is focusing on minerals; Lasaga (1997) has several chapters devoted to diffusion in minerals and to geochemical applications; Ghez (1988) is focusing on diffusion from the point of view of a crystal grower: perhaps it is the most stimulating reading for a mineralogist interested in mineral transformations in rocks. Finally, two books present a deep insight in transport and diffusion-reaction phenomena: Bird et al. (1960) and De Groot & Mazur (1962). For its high symmetry, garnet has a peculiar place in petrological works: here we would like to mention the interesting paper by Loomis (1978) on diffusion in garnet.

Empirical laws of diffusion

In the following the focus is on isothermal, isobaric diffusion of non-charged species, in absence of external forces and without chemical reactions, so that the driving forces are gradients of chemical potentials only. The derivation follows that by Haase (1969), but the equations will be written in a one dimension space, although the x component of vectors will be indicated with a bar, for notation clarity and to remember their origin.

The first quantitative formulation of the experiments by Thomas Graham, on diffusion in gas and liquids, between 1830-1850, is due to Fick in 1855. The ingredients are:

  • the flux Ji of component i: it is the quantity of i passing through a unit area of a reference surface during unit time;

  • the concentration gradient of i, measured in the direction perpendicular to the reference surface.

To appreciate the definition it is useful to consider that mass transport occurs by movements of atoms having different mass, volume and local averaged velocity. The local velocity, measured in a stationary reference frame, is an average over a volume large compared to the atomic dimensions, while very small compared to the measurable dimensions. 
formula
(37)
The weighted average flow velocity is 
formula
(38)
at can be the mole fraction of component i and examples of weights at will be seen in the following. The diffusion motion of i is now described relative to the average flow velocity. This flow defines a frame of reference in which the diffusive motion of each species is gauged (Rosenberger, 1979). Thus the diffusion velocity of i with respect to  
formula
is defined as 
formula
(39)
The total flux of a component relative to a stationary frame (the laboratory frame) is the product of a concentration, for instance mass density, times a velocity: 
formula
(40)
The quantity  
formula
is called the diffusion current density of species k or the diffusion flux of species k. It can be verified that 
formula
(41)
A reference velocity appropriate in diffusion experiments is the mean volume velocity of a volume element: 
formula
(42)
ck, V, and  
formula
are respectively the molarity, the partial molar volume and the average velocity of species k. The flux of species k is 
formula
(43)
From experimental investigations it is found that 
formula
(44)
where  
formula
and c1 are taken as dependent quantities. The Dik are the (n - 1)2 diffusion coefficients. They are functions of temperaure, pressure, concentrations. They do not depend on concentration gradients.

The local mass balance equation is 

formula
(45)
In Equation 45  
formula
is the partial derivative with respect to time at fixed position. The convective term 
formula
can be neglected in most diffusion experiments, so that inserting Equation 44 in Equation 45 we obtain: 
formula
(46)
Equations 44 and 46 are the generalisation of Fick's first and second laws respectively.

Thermodynamic theory

In the case of isothermal, isobaric diffusion of non-charged species, in absence of external forces and without chemical reactions, the local entropy production is 

formula
(47)
In Equation 47 the generalised fluxes Jk are referred to any reference velocity. 
formula
(48)
 
formula
(49)
In Equation 48 both the n generalized fluxes and forces are linearly dependent. The dependent quantities can be eliminated using, for example, the relationship: 
formula
(50)
Then we obtain 
formula
(51)
and from 
formula
(52)
 
formula
is the diffusion flux of species k with respect to the reference velocity  
formula
. The last expression of the local entropy contains only independent fluxes and forces. In terms of the n - 1 independent forces Xk given by Equation 49 and of the phenomenological coefficients Lik, it is: 
formula
(53)
The last relationship can be transformed expressing Xk in terms of concentration gradients: 
formula
(54)
where 
formula
(55)
In order to compare Equation 54 with Fick's first law,  
formula
must be expressed in terms of  
formula
: 
formula
(56)
By substituting in Equation 56 the expression of  
formula
given by Equation 44 we obtain 
formula
(57)
By comparison of relationship (57) with (54) it is seen that the phenomenological equations lead to the generalised Fick's law.

The Onsager reciprocity law Lj = Lji, (i, j = 2, 3, …, n) allows to reduce the number of the independent diffusion coefficients to -21 n(n - 1).

Diffusion in garnet

Garnet is one of the most studied minerals. In particular, Ca, Mg, Fe, Mn diffusion coefficients have been measured as a function of temperature and pressure, although with different degres of accuracy. The most recent measurements are reported in a series of works by Ganguly, Chakraborty and co-workers (Chakraborty & Ganguly, 1991, 1992). In a more recent one (Ganguly et al., 1998), the measurements have been tabulated as self-diffusion coefficients. These are defined in terms of the rate of transfer of components across a section fixed, so that no bulk flow occurs through it. This makes easier to interpret diffusion in terms of random molecular motion. From the self-diffusion coefficients it is possible to calculate the interdiffusion coefficients on the basis of the mean-field theory by Lasaga (1979). The Lasaga model takes into account the ionic nature of the diffusing cations in garnet.

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Acknowledgements

This chapter receives benefit from the criticism and suggestions by T. Nishiyama. We also acknowledge R. Compagnoni, F. Abbona and D. Aquilano, for critical comments and discussions.

1
Although in the paper there are no further “technical” indications about the retention/loss factors, we believe that they are a parameterisation of the boundary conditions for the diffusion equation in garnet, in such a way to fix the relative fluxes of each cation in or out the resorbing garnet.
2
A similar approach has been applied by Rebay & Powell (2002) to study eclogite facies metatroctolites (a gabbro originally consisting of the bimineralic assemblage plagioclase + olivine) from a variety of Western Alps localities, that preserve the original igneous texture. An equilibrium view of the mineral assemblages within plagioclase and olivine microdomains has been investigated, assuming local equilibrium at an appropriate scale (smaller than the original grain scale), and considering two different EBC for the microdomains.
3
When the equilibrium sets in, the grossular end-member component decreases with increasing temperature and pressure, as shown in Figure 17. A similar pattern is obtained when plagioclase has the equilibrium composition but Ca is segregated in garnet.
4
The system phases and phase components are:plagioclase (Ab, Or, An); K-feldspar (Ab, Or, An); biotite (Ann, Eas, Phl, Mn-bt); phengite (Mg-cel, Fe-cel, Ms); garnet (Alm, Prp, Grs, Sps); Jd; Zo; Qtz; H2O.
5
Let us consider a scalar flow Js = LssXs + LsvXv and a vectorial flow Jv = LvsXs + LvvXv. Xs is a scalar force while Xv is a vectorial one. is a scalar relating the scalar force Xs to the scalar flow Js. Lsv must be a vector relating the vectorial force Xv to the scalar flow Lvs, relating a scalar force to a vectorial flow, must be a vector as well. This kind of coupling cannot occur in isotropic systems where a reversal of the sign of all coordinates axes must leave all the phenomenological coefficients invariant. The scalar coefficients Lss and Lvv are invariant but Lsv and Lvs would change sign upon inversion of the coordinate system. So it must be = 0 and Lvs = 0. A rigorous presentation can be found in Eu (1992).
Copyright permissions Figures 1, 10 and 11 are reprinted from American MineraIogist, Vol. 87, Carlson, W.D.: Scales of disequilibrium and rates of equilibration during metamorphism, pp. 185-204, copyright 2002, with permission from the Mineralogical Society of America. Figures 2, 3 and 4 are reprinted from Geochimica et Cosmochimica Acta, Vol. 41, Joensten, R.: Evolution of mineral assemblage zoning in diffusion metasomatism, pp. 649-670, copyright 1977, with permission from Elsevier. Figures 5, 6, 7, 8 and 9 are reprinted from JournaI of PetroIogy, Vol. 41, Ashworth, J.R. & Chambers, A.D.: Symplectic reaction in olivine and the controls of intergrowth spacing in symplectites, pp. 285-304, copyright 2000, with permission from Oxford University Press. Figure 16 is reprinted from Journal of Metamorphic Geology, Vol. 19, Bruno, M., Compagnoni, R. & Rubbo, M.: The ultra-high pressure coronitic and pseudomorphous reactions in a metagranodiorite from the Brossasco-Isasca Unit, Dora Maira Massif, Western Italian Alps: A petrographic study and equilibrium thermodynamic modelling, pp. 33-43, copyright 2001, with permission from Blackwell Publishing. Figures 19 and 20 are reprinted from Contributions to MineraIogy and PetroIogy, Vol. 137, Rubbo, M., Borghi, A. & Compagnoni, R.: Thermodynamic analysis of garnet growth zoning in eclogitised granodiorite from M. Mucrone, Sesia Zone, Western Alps, pp. 289-303, copyright 1999, with permission from Springer.

Figures & Tables

Fig. 1.

Schematic chemical potential (&) gradients around a growing crystal for elements with very slow, slow, medium, and fast rates of intergranular diffusion. From Carlson (2002), modified.

Fig. 1.

Schematic chemical potential (&) gradients around a growing crystal for elements with very slow, slow, medium, and fast rates of intergranular diffusion. From Carlson (2002), modified.

Fig. 2.

Stable minerals in the system CaO - AlO15 - SiO2 - CO2: The minerals in (b) are stable at higher temperature and lower pressure than in (a). Block diagram in (b) shows the sequence of mineral layers that results from reaction between blocks of calcite and anorthite placed in contact in (a). From Joensten (1977), modified.

Fig. 2.

Stable minerals in the system CaO - AlO15 - SiO2 - CO2: The minerals in (b) are stable at higher temperature and lower pressure than in (a). Block diagram in (b) shows the sequence of mineral layers that results from reaction between blocks of calcite and anorthite placed in contact in (a). From Joensten (1977), modified.

Fig. 3.

Projection of saturation surface along µCaO axis, onto µAlO1µSio2 plane. Chemical potentials for 1027 °C and 350 bars. From Joensten (1977), modified.

Fig. 3.

Projection of saturation surface along µCaO axis, onto µAlO1µSio2 plane. Chemical potentials for 1027 °C and 350 bars. From Joensten (1977), modified.

Fig. 4.

Closed cycle of diffusion controlled reactions exchanging CaO, AlO15 and SiO2 between the isothermal-isobaric invariant assemblages C + W + G and W+ G + A, resulting in growth of the W + G layer at the expense of C and A. From Joensten (1977), modified.

Fig. 4.

Closed cycle of diffusion controlled reactions exchanging CaO, AlO15 and SiO2 between the isothermal-isobaric invariant assemblages C + W + G and W+ G + A, resulting in growth of the W + G layer at the expense of C and A. From Joensten (1977), modified.

Fig. 8.

Geometry of the growing symplectite: λ is the recurrence distance of the lamellae of α and β phases; δ is the thickness of the diffusion layer between lamellae and the reactant minerals of the matrix. Pα is the volume proportion of α. From Ashworth & Chambers (2000), modified.

Fig. 8.

Geometry of the growing symplectite: λ is the recurrence distance of the lamellae of α and β phases; δ is the thickness of the diffusion layer between lamellae and the reactant minerals of the matrix. Pα is the volume proportion of α. From Ashworth & Chambers (2000), modified.

Fig. 18.

SEM backscattered image of the Monte Mucrone metagranodiorite. Biotite is partially replaced by phengite, and a garnet corona developed between biotite and plagioclase, now replaced by a polycrystalline aggregate of Zo + Jd(Ab) + Qtz.

Fig. 18.

SEM backscattered image of the Monte Mucrone metagranodiorite. Biotite is partially replaced by phengite, and a garnet corona developed between biotite and plagioclase, now replaced by a polycrystalline aggregate of Zo + Jd(Ab) + Qtz.

Fig. 19.

Measured compositional pattern. Spot analyses are 2 μm spaced. Bt, Core and Pl correspond to the garnet-biotite interface, the garnet core and the garnet-plagioclase interface, respectively. From Rubbo et al. (1999), modified.

Fig. 19.

Measured compositional pattern. Spot analyses are 2 μm spaced. Bt, Core and Pl correspond to the garnet-biotite interface, the garnet core and the garnet-plagioclase interface, respectively. From Rubbo et al. (1999), modified.

Table 1.

Initial phase compositions for the three sites considered. The moles of phase components are normalised to 20 moles of cations.

PhasePhase componentsSite ASite BSite C
BtAnn0.2200.2200.088
Phl0.1500.1500.060
Eas0.1200.1200.048
Mn-bt0.0100.0100.004
Qtz14.00
KfsAb0.0300.600
Or0.1602.390
An0.0100.010
PlAb2.448
Or0.072
An1.080
PhasePhase componentsSite ASite BSite C
BtAnn0.2200.2200.088
Phl0.1500.1500.060
Eas0.1200.1200.048
Mn-bt0.0100.0100.004
Qtz14.00
KfsAb0.0300.600
Or0.1602.390
An0.0100.010
PlAb2.448
Or0.072
An1.080
Table 2.

Initial moles of end members in the EBC.

AnnPhlEasMn-btAbOrAnQtz
0.18620.141140.172.66−30.150.600.00.5
AnnPhlEasMn-btAbOrAnQtz
0.18620.141140.172.66−30.150.600.00.5

Contents

GeoRef

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