Coronitic reactions: Constraints to element diffusion during UHP metamorphism

Published:January 01, 2003
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, nonequilibrium 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, nonequilibrium 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).
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 nonequilibrium 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 twomineral 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 CaCO_{3} adjacent to anorthite CaAl_{2}Si_{2}O_{8}. 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  AlO_{15}  SiO_{2}  CO_{2} are seen in Figure 2. When temperature increases calcite (C) and anorthite (A) are incompatible and react with one another (Fig. 2).
The assemblage CA 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 CaSiO_{3} wollastonite (W) and Ca_{2}Al_{2}SiO_{7} 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 dissolutionprecipitation 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):
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 GibbsDuhem equation for every mineral constrains the components' chemical potentials in the solids and in the intergranular medium through the equilibrium condition µ_{i}^{mineral} = µ_{i}^{medium};
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 steadystate 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.
 Massbalance 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, AlO_{1}_{5}, SiO_{2} while it is supposed that no CO_{2} gradients exist. CO_{2} 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. where is the number of mole of component i consumed (if it is negative), or evolved in the reaction at the contact is the amount in moles of phase ϕ involved in the reaction and the stoichiometric coefficient i in the formula of phase ϕ.So for instance, for the component CaO, the equation at the boundary CW + G reads: and at the boundary The opposite sign in the equations is a consequence of the closed system assumption . There are 4 equations at the boundary C  W + G and 3 at the boundary W + GA, in this example.
 Steady diffusion equations
 Within the layers no chemical reaction occurs and
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:
 Flux ratio equations. Within the layer W + G the fluxes are related to the chemical potential gradients by: Note that cross coupling coefficients (L_{ij} = 0) are neglected and L_{ii} 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 L_{ii} are not known it is usual to write flux ratio equations (in the following J_{S}^{W}O^{G} is the common divisor):
 GibbsDuhem 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 . For the reaction band W + G: 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 extentofreaction 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 CW + G: One extent of reaction is fixed: The steady diffusion equations lead to: The following equations describe the reaction at the interface W + G A: 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: After substituting in the appropriate equations the values obtained from the system (1011), and for the tentatively assigned values the solution (29) of the linear system (28) allows to write down the reactions (Joensten, 1997):
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 CO_{2} 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 endmembers, 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 diffusionreaction 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.
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. 57) 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, 45 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 olivineclinopyroxene cumulates (600 m) passing upwards into gabbroic olivineclinopyroxeneplagioclase cumulates (1800 m) and then into a series of plagioclaseamphibole cumulates (400 m). The symplectites are found in olivinechromo spinel cumulates (Lower zone) as thin platelets of clinopyroxene and magnetite parallel to (100) of olivine when it is less ferroan than Fo_{74}. 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:
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.
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).
To calculate ∆G_{dif} we must know the diffusive flux, J_{i}, 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 J_{i} 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 P_{α}. 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
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 ∆G_{dif} 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 −ΔG_{gb}. Finally −ΔG_{ext} increases while −ΔG_{dif} 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(−ΔG_{diff}− ΔG_{ext}), whose expression is obtained using Equations 33, 47, 52:
Differentiating in respect to V the righthand side member of Equation 53 and equating the result to zero, we obtain: The expression of is obtained by differentiating Equation 49 at constant ΔG: Substituting Equation 55 in Equation 54 and multiplying by V the resulting expression, we obtain: So the maximum occurs when The relationship (57) between growth rate and spacing λ, holds when the rate of energy dissipation is maximal. It can be written as: The condition of maximum in terms of AG is (Ashworth & Chambers, 2000): which gives the relationship equivalent to Equation 58: in both cases: ΔG_{ext} = 0 and ΔG_{ext} ≠ 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 L_{k} 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 L_{Si}δ. In particular, from geological information it is estimated V >6 × 10^{−16} m^{3}s^{−1}; from experiments on olivine oxidation γ< 0.3 Jm^{−2}; it is measured λ≈ 4 μm. Finally, considering the term containing L_{Si} as the only one contributing to b, the sum in (52) reduces to: Identifying α with magnetite, . From Equation 57: being as results from the measured Pα ≈ 0.25 and the concentration of Si in olivine. There are experimental indications that the calculated value of L_{Si}δ occurs in the temperature range 8001000 °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 fluidundersaturated 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 twostage 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.
 1.
 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.
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.
The simulation is iterative. Given an initial temperature T_{0} 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 t_{f} 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 (D_{Al}_{ef}) by equating the measured length scale for intergranular diffusion in the coronal textures with the characteristic diffusion distance, travelled by diffusing atoms:
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 T_{0} (655686 °C); all other parameters are identical in all 14 fits. The estimated intergranular diffusivity for Al, resulting from this procedure, is: This equation is only applicable to hydrous but fluid undersaturated systems and in the range of temperature encompassed in these models (500650 °C).The ultrahigh pressure coronitic reactions in a metagranodiorite
Geology and petrography
Metagranodiorite samples from the BrossascoIsasca Unit (Biino & Compagnoni, 1992; Bruno et al., 2001), DoraMaira Massif, western Alps, show pseudomorphous and coronitic textures where igneous minerals were partially replaced by ultrahigh pressure (UHP) metamorphic assemblages. The BrossascoIsasca 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, Kfeldspar, biotite and accessory apatite, zircon and a Tirich 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 Kfeldspar (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 + Kfeldspar to a pseudomorph (Fig. 12). Zoisite and kyanite are pure, and Kfeldspar is low in albite (Or_{90}Ab_{10}). Napyroxene is a solid solution of jadeite, CaTschermak and CaEskola (Bruno et al., 2002), but usually it is partially replaced by retrograde albite or oligoclase. At the original igneous biotitequartz contact (Site A), a single continuous corona of weakly zoned garnet, with composition Alm_{7678}Prp_{2123}Grs_{12}Sps_{12}, develops (Fig. 13). Usually, garnet is rimmed by a retrograde biotite. Between biotite (partially replaced by phengite I) and Kfeldspar (Site B), the following composite corona is formed (Fig. 14):
a continuous corona of weakly zoned garnet (Alm_{7880}Prp_{1517}Grs_{3}Sps_{2});
a continuous corona of garnet (Alm_{78–80}Prp_{1719}Grs_{23}Sps_{01}) with vermicular quartz inclusions, elongated perpendicular to the corona;
a continuous corona of a quartzphengite (PhII) symplectite in continuity with the garnetquartz symplectite.
a corona of garnet (Grs_{4950}Alm_{42}Prp_{78}Sps_{01}) plus quartz;
a composite corona of idioblastic garnet (Grs_{62–84}Alm_{1632}Prp_{06}) 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 BrossascoIsasca 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.
Phase  Phase components  Site A  Site B  Site C 

Bt  Ann  0.220  0.220  0.088 
Phl  0.150  0.150  0.060  
Eas  0.120  0.120  0.048  
Mnbt  0.010  0.010  0.004  
Qtz  14.00  
Kfs  Ab  0.030  0.600  
Or  0.160  2.390  
An  0.010  0.010  
Pl  Ab  2.448  
Or  0.072  
An  1.080 
Phase  Phase components  Site A  Site B  Site C 

Bt  Ann  0.220  0.220  0.088 
Phl  0.150  0.150  0.060  
Eas  0.120  0.120  0.048  
Mnbt  0.010  0.010  0.004  
Qtz  14.00  
Kfs  Ab  0.030  0.600  
Or  0.160  2.390  
An  0.010  0.010  
Pl  Ab  2.448  
Or  0.072  
An  1.080 
The equilibrium calculations have been performed by minimising the Lagrangian,
at constant temperature and pressure (Smith & Missen, 1982). G is the Gibbs function: 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, μ_{i} is the chemical potential of each endmember component in each phase, and n_{i} are their mole numbers which are constrained by the mass balance equation (An  b = 0) and nonnegativity conditions (n_{i} > 0); ñ is the transpose of n.The abundance and composition of the phases have been calculated along a PT trajectory described by the following equation:
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 PT region between 3 and 40 kbar, and 420 and 830 °C.Thermodynamic modelling was undertaken in the KNCFMnMASH system with biotite (Bt), plagioclase (Pl), Kfeldspar (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 Mnbiotite (Mnbt) for biotite; albite (Ab), anorthite (An) and orthoclase (Or) for feldspar; muscovite (Ms), Feceladonite (Fecel) and Mgceladonite (Mgcel) for potassic white mica; almandine (Alm), grossular (Grs), pyrope (Prp), and spessartine (Sps) for garnet. The fluid is supposed to be pure H_{2}O.
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: where with n^{0}_{Bt} being the initial amount in moles of igneous biotite, X^{0}_{Eas} the initial molar fraction of eastonite in igneous biotite, X_{Eas} the final molar fraction of eastonite in metamorphic biotite and X_{Ms} the final molar fraction of muscovite. Reaction (69) stops when eastonite is consumed (X_{Eas} = 0). Instead, in site C the reactions producing garnet are (69) and:
On a thermodynamic basis it has been assessed that, for peculiar bulk chemical compositions, biotite is stable all along the calculated PT 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 nettransfer 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 waterconstant 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 (Alm_{7678}Prp_{2123}Grs_{12}Sps_{12}) matches the calculated at 24 kbar. Thermodynamic calculations also show that the Brossasco metagranodiorite experienced no further reactions for the PT range between 28 kbar and 40 kbar and 690 to 830 °C (Fig. 16).
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 PT 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 reequilibration. 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, reequilibration 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 Cafree phases but garnet contains Ca; zoisite grew at the expense of the original plagioclase (site P) but hydroxyl groups diffused from closeby 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 alkalirich 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 SesiaLanzo 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 = 500600 °C and P = 1620 kbar (Compagnoni, 1977; Lardeaux et al., 1982; Droop et al., 1990).
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 Xray compositional maps shows islands of nearly concentric isoconcentration 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 BrossascoIsasca 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 PT 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 PT 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 (T_{0}) and pressure (P_{0}). 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.
Temperature and pressure are iteratively changed by small increments, δT and δP, along an elliptic PT 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.
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: n_{k} stands for the vector whose elements are (n^{θ}_{i})_{k}, where θ runs on the phases, i runs over the phase components, and k is associated to the iteration^{4}. M_{ϕ}(i.e. M_{g} for the garnet and M_{p} for plagioclase) is the integral of the growth/dissolution rate of the crystal, at temperature T_{k} and pressure P_{k} over a time Δt. If Mϕ is positive, the phase ϕ grows, whereas it resorbs if Mϕ is negative.To generate a spherical crystal of garnet, the moles M_{g} 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.
M_{ϕ} is calculated as a balance between the number of moles added, E_{ϕ}(T_{k}, P_{k}, n_{k}), and subtracted, (T_{k}− δT, P_{k} δP), to phase ϕ. E_{ϕ}(i.e. E_{g} for the garnet) is the number of moles of the outer shell of phase ϕ having the equilibrium composition at temperature T_{k}, pressure P_{k}, when the moles of endmembers making the EBC are n_{k}.
E_{ϕ} 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).
N_{ϕ} is a function of Arrhenius type. It is calculated at (T_{k}  5T, P_{k} − δP) and it is the number of moles of phase ϕ to be resorbed at T_{k}, P_{k}.
The explicit expressions of N_{ϕ}used are:
T_{0} = 608 K is the lowest temperature of the considered prograde trajectory and T_{k} the temperature of the k^{th} iteration. To increase the input of Ca in the EBC, due to the dissolution of plagioclase, N_{p} is made increasing with temperature. Conversely, N_{g} 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 reequilibration 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 α_{p}, α_{g}, β_{p}, β_{g}: so, they are determined by trial and error until the calculated concentrations satisfactorily fit the measured profiles. The optimal values are: α_{g} = 0.12, β_{g} = 14, α_{p} = 2.5, β_{p} = 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:
The MMM garnet records a typical clockwise PT 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: D_{Ca} < D_{Mg} < D_{Fe} = D_{Mn}.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 PT 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 PT 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 pV 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.
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. 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 ccomponent phase is:
h_{k} is the partial molar enthalpy of species k and d_{e}n_{k}, d_{t}n_{k} 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 T^{a}, p^{a} and T^{E}, p^{E}, respectively, matter can be exchanged between phases only (d_{e}n_{k} = 0).
From Equation 3 written for every phase we obtain: The heat can be partitioned: that exchanged with the surroundings and that within the system between the two phases: The heat exchanged between the whole system and the surroundings is d_{e}Q^{α} + d_{e}Q^{β}. The variation of the whole system's internal energy is: The energy conservation requires: and thus substituting (5) and (6) in (7) and (8) and the result in (10) we find: 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: Only if (Haase, 1969). In the case the phases cannot exchange matter between them, it is: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 (d_{e}S) and a nonnegative source term due to irreversible processes occurring within the system (d_{i}S), that is
If the system experiences only reversible modifications, then d_{i}S = 0.Let's consider an isolated system composed of two phases αand β. For the whole system the following relation holds:
Within each phase, the production of entropy is nonnegative: A situation such that dS > 0 with 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 T^{α} and T^{β}. The system entropy is the sum
Every phase exchanges heat with the surroundings and with the other phase: As a consequence of the first principle, we have: and then: These contributions can be grouped in the following way: According to experience, if T^{α}< T^{β}then d_{i} Q^{α}>0 and entropy is produced by the internal process. When the temperature is uniform then d_{i}S = 0. The rate of production of entropy is: Let us observe that is the product of the rate (or flux) with the function of state , 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 T^{α}, T^{β}, pressure p^{α}, p^{β}, and chemical potential
of the k component. Let suppose that between the two phases an exchange of matter occurs. For the ccomponent subsystem (dropping the superscipts for the moment) the first principle reads: The Gibbs equation (obtained combining the first and second principles) is: Now dn_{k} ≡ d_{i}n_{k} describes the moles variation for the exchange within the system, between the phases. The entropy balance dS = dS^{α} + dS^{β} (Eqn. 13), is calculated by Equations 23 and 22: The rate of entropy production is calculated by introducing Equation 11: Only if and 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
are the stoichiometric coefficients of the reacting species. Introducing the affinity (A) and the reaction rate (v) the entropy balance reads 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
is a product of a rate or generalised flux (J_{k}) and a state function which is a generalised affinity or driving force(X_{k}). In the example on the chemical reaction J_{K} = v and .The rate of entropy production
has therefore the following structure: At equilibrium both fluxes and driving forces are zero for every irreversible process: 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
This assumption implies that if one driving force, say X_{1}, is zero, the flow J_{1} ≠ 0, being driven by X_{2}. 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: With reference to Equation 32 it can be shown that, being positive for all positive and negative values of X_{1}, X_{2}, except X_{1} = X_{2} = 0, in which case = 0, the following conditions hold: To generalise using the mathematical language, is a definite positive quadratic form.A second central point is the theorem by Onsager, stating that the coupling coefficients are equal:
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 X_{1}, X_{2}, is kept in a state with fixed say X_{1}, and minimum entropy production, the flux J_{2}, conjugate to the nonfixed force, vanishes. Deriving Equation 32 with respect to X_{2}, keeping constant X_{1}, using Equation 35, one obtains:
Conversely if X_{1} is fixed and J_{2} = 0 the entropy production per unit time is minimal. Being > 0, the condition (36) identifies a minimum.To obtain (36) the L_{ij} are assumed constant.
Up to now, examples of irreversible processes in socalled 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 diffusionreaction 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 noncharged 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 18301850, is due to Fick in 1855. The ingredients are:
the flux J_{i} 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.
The local mass balance equation is
In Equation 45 is the partial derivative with respect to time at fixed position. The convective term can be neglected in most diffusion experiments, so that inserting Equation 44 in Equation 45 we obtain: 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 noncharged species, in absence of external forces and without chemical reactions, the local entropy production is
In Equation 47 the generalised fluxes J_{k} are referred to any reference velocity. In Equation 48 both the n generalized fluxes and forces are linearly dependent. The dependent quantities can be eliminated using, for example, the relationship: Then we obtain and from is the diffusion flux of species k with respect to the reference velocity . The last expression of the local entropy contains only independent fluxes and forces. In terms of the n  1 independent forces X_{k} given by Equation 49 and of the phenomenological coefficients L_{ik}, it is: The last relationship can be transformed expressing X_{k} in terms of concentration gradients: where In order to compare Equation 54 with Fick's first law, must be expressed in terms of : By substituting in Equation 56 the expression of given by Equation 44 we obtain 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 = L_{ji}, (i, j = 2, 3, …, n) allows to reduce the number of the independent diffusion coefficients to _{2}^{1} 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 coworkers (Chakraborty & Ganguly, 1991, 1992). In a more recent one (Ganguly et al., 1998), the measurements have been tabulated as selfdiffusion 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 selfdiffusion coefficients it is possible to calculate the interdiffusion coefficients on the basis of the meanfield theory by Lasaga (1979). The Lasaga model takes into account the ionic nature of the diffusing cations in garnet.
References
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.
Figures & Tables
Phase  Phase components  Site A  Site B  Site C 

Bt  Ann  0.220  0.220  0.088 
Phl  0.150  0.150  0.060  
Eas  0.120  0.120  0.048  
Mnbt  0.010  0.010  0.004  
Qtz  14.00  
Kfs  Ab  0.030  0.600  
Or  0.160  2.390  
An  0.010  0.010  
Pl  Ab  2.448  
Or  0.072  
An  1.080 
Phase  Phase components  Site A  Site B  Site C 

Bt  Ann  0.220  0.220  0.088 
Phl  0.150  0.150  0.060  
Eas  0.120  0.120  0.048  
Mnbt  0.010  0.010  0.004  
Qtz  14.00  
Kfs  Ab  0.030  0.600  
Or  0.160  2.390  
An  0.010  0.010  
Pl  Ab  2.448  
Or  0.072  
An  1.080 
Contents
Ultrahigh Pressure Metamorphism
This is the first volume in this series dealing with a petrological subject and contains the contributions of the lectures given at the 5th School of the European Mineralogical Union (EMU) on “Ultrahigh Pressure Metamorphism” held in Budapest from 21 to 25 July 2003. The topic of UHPM was selected because this extreme type of metamorphism, initially considered as a petrographic oddity by the geologic community, has now become recognised as a normal feature of continental plate collisional orogens and important to understanding just how deep the upper part of the continental lithosphere can subduct. We note that this School took place just twenty years from the first report by Christian Chopin of coesite in exposed orogenic metamorphic rocks of the continental crust. The lectures given at this school benefited by the scientific results of the research promoted by the ILP Task Groups III6 and III8, active on UHPM from 1994 to 1998 and 1999 to 2004, respectively, and published in a number of monographs and special issues of international journals. It is our strong belief that this petrologic topic should be recognised to be of paramount importance in the education of students and young researchers in Earth Science.