Determination of reaction kinetics using grain size: An application for metamorphic zircon growth

The reaction kinetics of metamorphic minerals can be subdivided into interface‐ and diffusion‐controlled kinetics. The discrimination of reaction kinetics is crucial for estimating reaction rates. Here, we propose a new and simple method for discriminating reaction kinetics. This method requires measuring only the initial and final grain sizes during growth. The reaction kinetics is inferred from different plotted arrays of initial vs. final grain sizes after the mineral growth. Using metamorphic zircon, we take detrital core sizes as the initial sizes and post‐metamorphic grain sizes as the final sizes. The application of the method to the subduction‐related high‐pressure Nagasaki metamorphic complex in Japan shows that this metamorphic zircon grew under interface‐controlled kinetics even at the relatively low temperature of 440°C. This method is potentially applicable to other minerals that have time‐markers, such as chemical zoning or internal structures that are captured at a given point in time during growth.


| INTRODUCTION
The kinetics of metamorphic reactions are subdivided into two regimes: interface-and diffusion-controlled. Interface-controlled kinetics imply that the rates of reaction at a mineral interface determine the net reaction rate, whereas the diffusion-controlled kinetics imply that the diffusion rate of the slowest chemical component controls the net reaction rate. Most metamorphic reactions involve dehydration or hydration. Therefore, the metamorphic reaction rates constrain the fluid flux. In addition, the growth rate of metamorphic minerals from progressive metamorphic reactions governs metamorphic rock texture.
Despite the efforts of previous studies of reaction kinetics, discriminating the type of kinetics is difficult, apart from a small number of cases. For example, Miyazaki (2015) determined the diffusion-controlled reaction kinetics of the garnet-forming reaction. In this case, garnet was produced with a clear diffusion halo, which is diagnostic of diffusion-controlled kinetics. However, garnets with clear diffusional haloes are rare. As such, reaction kinetics are usually assumed a priori when determining the reaction rates and fluid fluxes as well as the evolutions of metamorphic textures. For example, interface-controlled kinetics have been assumed for field measurements of high-temperature bulk reaction rates (Baxter & DePaolo, 2002a, 2002b and for the assessment of nonlinear reaction kinetics in the parameterization of metamorphic fluid flow (Zhao & Skelton, 2014). Diffusion-controlled metamorphic reactions have been assumed to be responsible for the textural evolutions of metamorphic rocks due to Ostwald ripening (Miyazaki, 1991, Miyazaki, 1996, diffusion-limited aggregation metamorphic rocks as well as predicted rock textures (the metamorphic grain size and the spatial distribution of minerals), depend strongly on reaction kinetics. Here, we present a new and simple method for the discrimination of reaction kinetics. We apply the method to the subduction-related high-pressure Nagasaki metamorphic complex of Japan.

| ME TH ODS
We propose a new method for discriminating reaction kinetics. The reaction kinetics of metamorphic minerals can be subdivided into interface-and diffusion-controlled types. The growth rate (da/dt) takes different forms depending on the reaction kinetics, which are written as follows (e.g. Kretz, 1994): where a is the grain size, and k 1 and k 2 are the kinetic coefficients for diffusion-and interface-controlled kinetics, respectively. The integrals of Equations 1 and 2 are, respectively, where a 0 is the initial grain size. Individual grains have the same kinetic coefficients at a given point in time, so the integration of approximately 1,000 km (Miyazaki, Ozaki, Saito, & Toshimitsu, 2016) and is a typical high-pressure and low-temperature, intermediate-type metamorphic complex. The Nishisonogi unit comprises pelitic schists with minor mafic schist, serpentinite, quartz schist and psammitic schist (Nishiyama, 1989;Nishiyama et al., 2017).
The protolith of the pelitic schist is a mudstone containing carbonaceous material (CM). The pelitic schist also contains detrital zircon grains. The detrital zircon U-Pb ages from psammitic schist (Kouchi et al., 2011) and the igneous zircon U-Pb ages of a jadeitite protolith (Mori, Orihashi, Miyamoto, Shigeno, & Nishiyama, 2011) indicate that the depositional age is <86 Ma. The metamorphic phengite K-Ar ages for the pelitic schists (Hattori & Shibata, 1982) and the 40 Ar/ 39 Ar ages of the phengite and biotite from jadeitite (Mori et al., 2011) show that metamorphism occurred between 85 and 60 Ma. The peak metamorphic conditions for the schists were correlated with those of the Sanbagawa metamorphic F I G U R E 1 Initial grain size (a 0 ) vs. grain size after growth (a) for (a) interface-and (b) diffusion-controlled kinetics. The initial size distribution was assumed to be Gaussian with a mean of 45 lm and standard deviation of 15 lm. The dashed line has a slope of one complex, which reached the garnet zone (Nishiyama, 1990). These conditions are 7.0-8.5 kbar and 440 AE 15°C in the garnet zone (Enami, Wallis, & Banno, 1994). The pelitic schist consists of quartz, albite, phengite, chlorite and garnet, with minor CM, zircon, apatite and titanite.
To observe the internal structure of the zircons and to measure their grain sizes, we obtained images from an optical microscope as well as cathodoluminescence (CL) and backscattered electron (BSE) images. CM was found in the zircon rims and in the matrix of the pelitic schists. We measured the degree of graphitization using

| Raman spectra and carbonaceous material thermometer
Metamorphic temperatures were estimated by RSCM (Figure 4) using the calibration of Aoya et al. (2010). The estimated temperature is approximately 440°C, which is consistent with the previously proposed metamorphic conditions (Nishiyama, 1990). The estimated temperature for the CM inclusions in the zircon rims is identical to those for the CM in the pelitic schist matrix (Figure 4). No differences were detected in the estimated temperatures among the samples analysed.

| Metamorphic zircon rim growth
The very fine-grained CM inclusions in the zircon rims and the matrix of the pelitic schist suggest that the zircon grains occluded the CM during rim growth. In addition, the CM inclusions in the zircon rims and the matrix of the pelitic schist experienced the same thermal histories. These observations strongly suggest that the zircon rims have a metamorphic origin and that the zircon cores are detrital. As described earlier, the protolith age is <86 Ma and metamorphism occurred between 85 and 60 Ma, thereby constraining the growth of metamorphic zircons to the period between 85 and 60 Ma.

| Discrimination of the growth kinetics of metamorphic zircon
We applied the discrimination method to the metamorphic zircon from the Nishisonogi unit of the Nagasaki metamorphic complex, Japan. We can take the measured a 0 from the detrital zircon size as the initial size. This is a unique feature of the discrimination of the growth kinetics using metamorphic zircon. The measured zircon sizes are scattered around the intrinsic average of the growth increments of each sample. We also evaluated the sectioning errors associated with measuring the grain sizes (see Appendix B). However, the scattering cannot be attributed to sectioning errors. Therefore, to compare an array of grain sizes with a calculated array, we introduce a stochastic dispersion of the kinetic coefficients k 1 and k 2 . The stochastic dispersion was introduced as a stochastic dispersion of the right-hand terms in Equations 3 and 4. The initial size distribution was assumed to be Gaussian, for which the mean and variance were assumed to be similar to those of the measured zircon core sizes. The kinetic coefficients were generated randomly between the maximum and minimum values and were adjusted to replicate the values of the means and the dispersions of the measured zircon sizes. When the maximum value is three times greater than the minimum value, the stochastic dispersion replicates similar values.
Although the stochastic dispersion results in scattered arrays, the differences in the arrays between diffusion and interface-controlled kinetics are still clear (Figure 6a F I G U R E 6 Initial grain size (a 0 ) vs. grain size after growth (a) for (a) interfacial-and (b) diffusion-controlled kinetics with dispersions of the kinetic coefficient. The initial size distribution was assumed to be Gaussian with a mean of 45 lm and standard deviation of 15 lm. The kinetic coefficients vary as a random stochastic value, where the maximum value is three times larger than the minimum value. The dashed line has a slope of one grains, the stochastic dispersion of the kinetic constant is reasonable for natural samples. Indeed, a growth rate dispersion depending on the surface condition of growing grains in the solution has been proposed as a real phenomenon (Srisanga et al., 2015).
The plotted arrays of a vs. a 0 for the Nagasaki metamorphic complex ( Metamorphic zircon growth requires a source of Zr. The source of Zr is somewhat problematic. However, it is possible that zircon grains smaller than the measured size (e.g. < 5-10 lm) dissolved through Ostwald ripening and supplied Zr for metamorphic zircon growth (e.g. Kawakami et al., 2013;Nemchin, Giannini, Bodorkos, & Oliver, 2001;Vavra, Schmid, & Gebauer, 1999). In addition, interface-controlled Ostwald ripening of zircon in a quartz + water system has been experimentally observed (Ayers, DeLaCruz, Miller, & Switzer, 2003). LSW theory (Lifshitz & Slyozov, 1961;Wargner, 1961) predicts that the mean diameter <D> increases with t 1/2 , although individual grain size D, which is much larger than the critical radius, increases with t for the interface-controlled kinetics at constant temperature and pressure conditions. Equation 4 shows that the individual grain size D increases with t at constant temperature and pressure conditions. Therefore, our results are consistent with the experimental results of Ayers et al. (2003).

| Implications
The reaction kinetics of metamorphic minerals cannot usually be directly determined from observations of natural samples (e.g. Miyazaki, 2015) due to the following factors: (1) the rarity of direct evidence of diffusion-controlled growth, such as diffusional haloes, and (2) the lack of a relationship between the interface-controlled growth and any specific spatial structure. Even if diffusion-controlled growth occurs, a diffusion halo will not form at very low degrees of super-saturation. However, the method presented in this paper could be used even in such situations, given that individual grains record reaction kinetics. The durations and kinetic coefficients do not affect the shapes of the arrays in our discrimination diagram (Figures 1a,b and 6a,b), which is solely dependent on the reaction kinetics.
The method presented in this paper is potentially applicable to

| CONCLUSION S
We proposed a new method for the discrimination of reaction kinetics using grain size that requires measuring only the initial and final grain sizes during growth. We applied this method to metamorphic zircon growth in the Nagasaki metamorphic complex of Japan. Detrital zircon can be used to constrain the initial sizes prior to metamorphic zircon growth, which is a unique feature for discriminating reaction kinetics using metamorphic zircon. The results suggest that the metamorphic zircon grew under conditions of sufficient fluid during high-pressure and low-temperature subduction zone metamorphism. The method presented in this paper is potentially applicable to other minerals that have time-markers at a given point in time, such as the chemical zoning in garnet or the internal structure of amphibole with both igneous and metamorphic origins.