Diffusion-controlled growth and degree of disequilibrium of garnet porphyroblasts: is diffusion-controlled growth of porphyroblasts common?
© Miyazaki. 2015
Received: 18 March 2015
Accepted: 12 August 2015
Published: 2 September 2015
Rate-limiting processes and the degree of disequilibrium during metamorphic mineral growth are key controls on the rate of dehydration and hydration in the Earth’s crust. This paper examines diffusion-controlled growth and the degree of disequilibrium of garnet porphyroblasts in the Tsukuba metamorphic rocks of central Japan. The analyzed porphyroblasts have irregular and branching morphologies with clear diffusional haloes, indicating that they grew in a diffusion-controlled regime. Mathematical analysis shows that the dominant wavelength of the interface of a garnet porphyroblast is dependent on the extent of supersaturation (Δζ), which is an index for the degree of disequilibrium. Using the calculated upper and lower limits of the dominant wavelength, the value of Δζ is estimated to be 0.05 × 10−1–0.16, which corresponds to a Gibbs free energy (ΔG r ) overstep of 0.9–27 kJ per mole of garnet (12 oxygen atoms) and a temperature overstep (ΔT) of 1.7–50 °C. Using the average value of the dominant wavelength, the following results are obtained: Δζ = 0.15 × 10−1, ΔG r = 2.7 kJ per mole of garnet, and ΔT = 5 °C. These values bring into question the importance of diffusion-controlled growth of garnet porphyroblasts, as highly irregular and branching garnet porphyroblasts are rare in most metamorphic belts. After significant overstepping for the nucleation of garnet, the garnet porphyroblasts grow at a high degree of disequilibrium. However, a high degree of disequilibrium under diffusion-controlled growth would be characterized by diffusional instability. The results indicate that garnet porphyroblasts that lack an irregular and branching morphology may grow at a high degree of disequilibrium under interface-controlled growth, provided they are set in a medium where the diffusion and supply of constituent elements are sufficient, such as a sufficient volume of metamorphic fluid.
KeywordsDiffusion-controlled growth Garnet Metamorphism Disequilibrium Porphyroblast
Rate-limiting processes and the degree of disequilibrium of a metamorphic reaction are crucial factors in the devolatilization and/or hydration of the Earth’s crust. The release and consumption of fluids, such as water-rich fluids and melts, have a critical influence on the evolution of mountain building and the transport of energy and materials in the crust. Many studies have assessed the rate-limiting processes for metamorphic reactions (e.g., Ague and Carlson 2013, Walther and Wood 1984). The following three types of rate-limiting processes have been recognized (e.g., Fisher 1978): 1) interface-controlled, 2) diffusion-controlled, and 3) heat-flow-controlled. The formation of garnet is one of the most common devolatilization reactions in the crust (e.g., Ague and Carlson 2013), and diffusion-controlled processes are widely applied to garnet-forming reactions, including the nucleation and growth of the garnet (e.g., Carlson 1989) and Ostwald ripening (Miyazaki 1991, 1996). Although Carlson (1989) and Miyazaki (1991) both assumed a diffusion-controlled process, the degree of disequilibrium differed in their models. In particular, the degree of disequilibrium for the nucleation and growth model is much higher than that for Ostwald ripening. Even if the rate-limiting processes were governed by diffusion, the degree of disequilibrium should directly affect the reaction rate and the patterns of metamorphic textures. The growth of a metamorphic mineral at a low degree of disequilibrium (i.e., a system close to equilibrium, implying a small overstep from the equilibrium temperature and a low degree of supersaturation for the metamorphic reaction) may result in an equigranular metamorphic texture because interfacial energy becomes an important control on crystal morphology and size. On the other hand, growth at a high degree of disequilibrium with a limited number of nuclei may result in a porphyroblastic texture. However, diffusion-controlled growth of a mineral at a high degree of disequilibrium is likely to involve diffusional instability and may lead to highly irregular and branched morphologies. In this paper, I examine the diffusional instability of garnet porphyroblasts (Tsukuba metamorphic rocks, central Japan) that grew in a diffusion-controlled regime.
Identification of a texture characteristic of diffusion-controlled growth
Mathematical analysis of diffusional instability
Equations 6 and 8 show that the dominant wavelength λ max (=2πR g /l max) can be directly extracted from the supersaturation Δζ. In reverse, this means that the observed dominant wavelength of the product interface should be linked to the supersaturation.
Although R c has a very small value, Eq. 12 implies that R l becomes very large for a large l.
Measurement of the garnet morphology
Equations 6 and 8 imply that the supersaturation Δζ for diffusion-controlled growth can be directly obtained from the dominant wavelength λ max of the interface of the growing product. Measurements of the dominant wavelength of garnet produced by diffusion-controlled growth were performed by measuring the radius of local curvature of the garnet interface (method 1), and by measuring the power spectra of the garnet interface (method 2).
Tsukuba metamorphic rock sample
The sample analyzed for this study was collected from the high-grade part of the Late Cretaceous low pressure–high temperature (LP–HT) Tsukuba metamorphic rocks that form an eastern extension of the Ryoke metamorphic complex. The locality of the sample (number Ma2-48a) was described by Miyazaki (1999). The protoliths of the Tsukuba metamorphic rocks were mudstones and sandstones of an accretionary complex of Late Jurassic to earliest Cretaceous age (Miyazaki et al. 1996). The Tsukuba metamorphic rocks were intruded by Late Cretaceous to earliest Paleogene granitic plutons. The metamorphic grade increases from southeast to northwest.
Based on the mineral assemblages, the pelitic metamorphic rocks can be assigned to the biotite and sillimanite zones. The mineral assemblage of the biotite zone rocks is biotite + muscovite + quartz + plagioclase ± andalusite, whereas the sillimanite zone rocks contain K-feldspar + sillimanite + biotite + quartz + plagioclase ± cordierite ± garnet. The boundary between the biotite and sillimanite zones is defined by the dehydration reaction of muscovite and quartz, which produces K-feldspar, sillimanite, and water. The sample analyzed for this study belongs to the high-grade part of the sillimanite zone and contains K-feldspar + garnet + cordierite + biotite + plagioclase + quartz + sillimanite. The average pressure–temperature (P–T) conditions of samples previously studied from the locality of the present sample are P = 0.34 ± 0.1 GPa and T = 642 ± 16 °C (Miyazaki 1999).
Based on digital maps showing the distribution of minerals, as compiled from X-ray intensity maps of Na, K, Mn, Fe, and Al, the mineral mode of the sample is 57 % quartz, 22 % K-feldspar, 8 % plagioclase, 6–7 % biotite, 4–5 % cordierite, 1–2 % garnet, and <1 % sillimanite (Miyazaki 2001). K-feldspar, biotite, plagioclase, and quartz are distributed randomly as matrix minerals with diameters of 50–60 μm and do not occur as porphyroblasts (Miyazaki 2001). Because of its low abundance in the sample, it is unclear if sillimanite also has a random distribution in the matrix.
Cordierites occur as porphyroblasts (diameter ≤1 mm) and exhibit irregular shapes. The porphyroblasts have a highly branched texture that may have been formed by diffusion-limited aggregation (DLA). The diffusion of Al may be rate-limiting for DLA-like cordierite porphyroblast growth (Miyazaki 2001). Comparisons of the morphologies and fractal dimensions of the cordierites with those of DLA indicate that formation of the cordierite porphyroblasts can be modeled as a DLA process (Miyazaki 2001). A finite-sized growth fluctuation, induced by a random distribution of matrix minerals, is amplified by the diffusional field and results in a DLA-like pattern of porphyroblast growth. Hence, diffusion-controlled growth within a random matrix field is essential for the formation of DLA-like patterns in cordierite porphyroblasts (Miyazaki 2001).
The chemical compositions of the constituent minerals were given by Miyazaki (1999). The garnet contains about 7 wt% MnO (73 % almandine, 16 % spessartine, 9 % pyrope, and 2 % grossular). Compositional zoning of MnO, FeO, MgO, and CaO was not detected (Fig. 9), but slightly higher MnO contents occur rarely at the tips on the outer rims of garnet porphyroblasts. Biotites are made up of the components phlogopite (41 %), eastonite (26 %), Ti-biotite (17 %), and muscovite (16 %). The end-member formulae of these components are phlogopite (KM3AlSi3O10[OH]2), eastonite (KM2Al2Si2O10[OH]2), Ti-biotite (KM1.5TiAl2Si2O10[OH]2), and muscovite (KAl3Si3O10[OH]2) (e.g., Ikeda 1990), where M represents the divariant cations of Mg, Fe, and Mn. The cordierites consist of hydrous cordierite (50 %) and Fe-cordierite (50 %). The K-feldspar contains 20 % of the albite component.
Garnet-forming reaction and reaction entropy
Mineral compositions and entropies at 642 °C (used in this paper)
S (J/mole K)
4Bt + 10Sil + 15.5Qtz = Grt + 4Kfs + 4.5Crd + 1.75 W
Using the standard-state entropy and heat capacity of the end-member minerals (Holland and Powell 2011) and assuming ideal mixing for garnet, biotite, and cordierite solid solutions, the reaction entropy at 642 °C becomes 542 J/K for the formation of 1 mole of garnet (12 oxygen atoms). The reaction entropy is also a function of temperature at constant pressure. Here, I assumed that the change in reaction entropy between 642 and 742 °C is not significantly different compared with the reaction entropy at 642 °C. For example, the reaction entropy is 541 J/K at 700 °C and 540 J/K at 742 °C.
Diffusional haloes around garnet porphyroblasts
The garnet-forming reaction (Eq. 16) shows sillimanite, biotite, and quartz as the reactants and garnet, cordierite, K-feldspar, and water as the products. As shown in Fig. 8, biotite is scarce in the area surrounding the garnet porphyroblasts. Biotite is the most important reactant that supplies Al, Fe, and Mg to a growing garnet crystal. Therefore, the observed depletion of biotite around garnet porphyroblasts provides direct evidence for diffusion-controlled growth. The depletion zone is recognized as diffusional haloes around garnet porphyroblasts. Sillimanite is also an important reactant that supplies Al to garnet, but it is scarce in the matrix. Because sillimanite is typically found farther away from the garnet, the depletion zone of sillimanite around the garnet is much more difficult to identify than the biotite depletion zone.
Cordierite and K-feldspar are products of the garnet-forming reaction. The cordierite crystals in the depleted zone are much smaller than the porphyroblastic cordierite. The number and size of K-feldspar grains in the depletion zone are also smaller than outside the depletion zone. These observations indicate that growth of cordierite and K-feldspar within the depleted zone was suppressed in comparison with growth of cordierite porphyroblasts and K-feldspar matrix minerals outside the depletion zone.
Dominant wavelength and supersaturation
Radius of curvature and dominant wavelength of garnet porphyroblast interfaces, and the estimated supersaturation (Δζ)
R g : radius (μm)
a: mid-range of radius of curvature (μm)
b: mid-range of dominant wavelength (μm)
c: average of the dominant wavelengths by 2 × a and b (μm)
Δζa lower bound calculated with value of b
Δζ*a mid-range value calculated with value of c
Δζ*a upper bound calculated with value of 2 × a
0.06 × 10−1
0.06 × 10−1
0.06 × 10−1
0.03 × 10−1
0.04 × 10−1
0.04 × 10−1
0.05 × 10–1c
0.15 × 10–1c
Gibbs free energy and temperature oversteps
Using Eq. 15, ΔG r , and the reaction entropy ΔS r of 542 J/K per mole of garnet for the garnet reaction, a temperature overstep ΔT of 1.7–50 °C is calculated using the lower and upper boundaries of the dominant wavelength. When the average dominant wavelength is used, a ΔT of 5 °C is calculated.
Estimates of the degree of disequilibrium for the garnet-forming reaction have been reported in previous studies. Pattison and Tinkham (2009) evaluated the temperature and Gibbs free energy oversteps for a garnet-forming reaction in the Nelson aureole (British Columbia) using discrepancies between phase-equilibrium modeling and field observations of the location of the garnet isograd. This approach yielded ΔT = 30 °C and ΔG r = 4.8 kJ per mole of garnet. The degree of disequilibrium for the Nelson aureole is similar to that calculated for the sample in this study. However, the garnet porphyroblasts in the Nelson aureole are euhedral rather than irregular and branching in the case of the garnets from the Tsukuba metamorphic rocks. Kinetic modeling based on interface-controlled nucleation and growth was used to explain the formation of the garnet in the Nelson aureole (Gaidies et al. 2011). On the other hand, Kelly et al. (2013b) used numerical modeling of diffusion-controlled nucleation and growth to reproduce the crystal sizes and spatial distributions of minerals in 13 porphyroblastic rocks. They calculated ΔT = 5–67 °C and ΔG r = 0.7–5.8 kJ per mole of garnet. Spear et al. (2014) also evaluated the degree of disequilibrium for the garnet-forming reaction. Using a combination of quartz in garnet barometry (QuiG) and thermodynamic modeling for garnet zone metamorphic rocks in eastern Vermont, they calculated a ΔT of 10 °C, a pressure overstep of 0.6 kbar, and a ΔG r of 2 kJ per mole of garnet. For the staurolite–kyanite zone in the same metamorphic terrain and for a blueschist sample from Sifnos, Greece, Spear et al. (2014) calculated a ΔT of 50 °C, a pressure overstep of 2–5 kbar, and a ΔG r of 10–18 kJ per mole of garnet. Although diverse methods have been used to infer the degree of disequilibrium for the garnet-forming reaction, the calculated values suggest that in all cases the garnet nucleated and grew after a significant overstep from equilibrium conditions.
Wilbur and Ague (2006) used Monte Carlo simulations of crystal growth to demonstrate the formation of irregular and branching morphologies in garnet porphyroblasts. Because this simulation assumed random walk of chemical species, the morphology was produced by diffusion-controlled growth. They determined that a minimum Gibbs free energy overstep of about 2 kJ per mole of garnet was needed to produce a branched morphology for silicate minerals (Wilbur and Ague 2006). This value is consistent with the Gibbs free energy overstep calculated in this paper. Spear and Daniel (2001) proposed an amoeba-like growth model for garnet porphyroblasts based on observations of heterogeneous chemical zoning. This model supports the development of an irregular and branching morphology in garnet porphyroblasts. Collectively, the results obtained using these different approaches suggest that the irregular and branching morphology of garnet porphyroblasts is indicative of diffusion-controlled growth.
Ostwald ripening and grain growth are not considered in this paper because the garnets occur as porphyroblasts, meaning that the size of the garnet crystals is much larger than the surrounding matrix minerals. Because grain growth associated with Ostwald ripening takes place due to the reduction of interfacial energy when the system is close to equilibrium, this mechanism should occur at a faster rate for smaller grain sizes. Hence, grain growth due to Ostwald ripening is only important for non-porphyroblastic minerals.
If the number of garnet crystals is much fewer than the number of reactant crystals, the evolution of garnet size and degree of supersaturation should follow path-A. For diffusion-controlled growth in this case, a small nucleation rate and large number of reactant crystals favor the growth of a small number of garnet crystals, thus producing unstable porphyroblastic garnet. Large garnet porphyroblasts with irregular and branching morphologies will form. Conversely, if the number of garnet crystals is much larger than the number of reactant crystals, the evolution of garnet size and degree of supersaturation should follow path-B. Due to a high nucleation rate and a small number of reactant crystals, many small garnets should grow rather than porphyroblasts. In addition, path-B promotes a stable garnet morphology, thus preventing the formation of branching and irregular textures.
Kelly et al. (2013b) proposed diffusion-controlled growth for the formation of garnet porphyroblasts whereas Gaidies et al. (2011) proposed interface-controlled growth. Based on the mean radii of garnet porphyroblasts measured in these studies, the mean diameters of garnet porphyroblasts range from 180 to 2400 μm (Fig. 13). Most diameters fall in the range of 1000 to 2000 μm. Therefore, diffusion-controlled growth of garnet porphyroblasts without the development of irregular and branching morphology is difficult to achieve. This finding contradicts the hypothesis that diffusion-controlled growth of garnet porphyroblasts dominates in nature, which is consistent with the rarity of irregular and branching garnet porphyroblasts in metamorphic belts.
Over a scale of 10 cm, temperature heterogeneities produced by the growth of a garnet porphyroblast will be quickly dissipated over a time interval significantly shorter than the long growth period of the porphyroblast. This reasoning indicates that a control on porphyroblast growth by heat flow is unlikely. Therefore, the results of this study suggest that garnet porphyroblasts lacking an irregular and branching morphology (common in many metamorphic belts) may grow at a high degree of disequilibrium under interfacial-controlled growth, provided they are set in a medium in which diffusion is rapid and the supply of elements from metamorphic fluids is sufficient.
Garnet porphyroblasts with an irregular morphology and a diffusional halo of reactants occur in the LP–HT Tsukuba metamorphic rocks of central Japan. The features observed suggest diffusion-controlled growth of the porphyroblasts.
Using an analysis of diffusional instability, the degree of disequilibrium can be determined by measuring the dominant wavelength of the interface of the garnet porphyroblasts. The results show that the supersaturation is 0.05 × 10−1–0.16. From the extent of supersaturation, the calculated overstep of Gibbs free energy from equilibrium is 0.9–27 kJ per mole of garnet (12 oxygen atoms), and the temperature overstep is 1.7–50 °C.
A high degree of disequilibrium for a garnet-forming reaction is expected during growth of a garnet porphyroblast after nucleation. The present results suggest that garnet porphyroblasts that lack irregular and branching morphologies, a typical feature of many metamorphic belts, may grow at a high degree of disequilibrium under interfacial-controlled growth, provided they are set in a medium where diffusion and the supply of elements is sufficient, such as a sufficient volume of metamorphic fluid.
fast Fourier transform
quartz in garnet barometry
The author would like to express his gratitude to Dr. Tetsuo Kawakami and Kazuhiko Ishii who were conveners of the Japan Geoscience Union session on “Deformed Rocks, Metamorphic Rocks and Tectonics” for encouraging me to submit this paper to PEPS. I also thank two anonymous reviewers for their constructive reviews, comments, and suggestions that helped to improve the manuscript.
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