Myrmekite and strain weakening in granitoid mylonites

At mid-crustal conditions, deformation of feldspar is mainly accomplished by a combination of fracturing, dissolution/precipitation and reaction-weakening mechanisms. In particular, K-feldspar is reaction-weakened by formation of strain-induced myrmekite a fine-grained symplectite of 10 plagioclase and quartz. Here we investigate with EBSD the microstructure of a granodiorite mylonite, developed at 420-460 °C during cooling of the Rieserferner pluton (Eastern Alps), to assess the microstructural processes and the role of weakening associated with myrmekite development. Our analysis shows that the crystallographic orientation of the plagioclase of pristine myrmekite was controlled by that of the replaced K-feldspar. Myrmekite nucleation resulted in both grain size 15 reduction and ordered phase mixing by heterogeneous nucleation of quartz and plagioclase. The fine grain size of sheared myrmekite promoted grain size-sensitive creep mechanisms including fluidassisted grain boundary sliding in plagioclase, coupled with heterogeneous nucleation of quartz within creep cavitation pores. Flow laws calculated for monomineralic quartz, feldspar, and quartz + plagioclase aggregates (sheared myrmekite), show that during mylonitization at 450 °C grain-size20 sensitive creep in sheared myrmekite accommodated strain rates several orders of magnitude higher than monomineralic quartz layers deforming by dislocation creep. Therefore, diffusion creep and grain size-sensitive processes contributed significantly to bulk rock weakening during mylonitization. Our results have implications for modelling the rheology of the mid-upper continental (felsic) crust.

Here we present a detailed analysis of myrmekite evolution, from the nucleation stage within Kfs to the development of sheared Plg + Qtz aggregates, and of the associated rheological weakening that resulted in strain localization in the mylonites of the Rieserferner granitoid pluton (Eastern Alps).In this pluton, ductile shear zones nucleated along joints that were locally filled with Qtz and Ep (epidote) veins during post-magmatic cooling (Ceccato et al., 2017;Ceccato and Pennacchioni, 2018).The progressive development of granodiorite mylonite was associated with consumption of Kfs by myrmekite leading to increasingly interconnected fine-grained Plg + Qtz layers.Microstructures of granodiorite mylonite have been analysed to characterize the following: (i) the process of myrmekite nucleation; (ii) the deformation mechanisms during myrmekite shearing and transition to Plg + Qtz aggregates; (iii) the deformation mechanisms of pure Qtz layers; and (v) the deformation mechanisms of Kfs porphyroclasts and of new Kfs grains during mylonitization.Furthermore, the application of mixed flow laws of the aforementioned deformation mechanisms for polymineralic aggregates allows for the degree of rheological weakening resulting from deformation of myrmekite to be quantified.The current paper represents a first attempt at quantitatively estimating, based on a robust microstructural analysis, the rheological effects of the development of myrmekite during mylonitization of granitoids, and at determining the deformation mechanisms in syn-kinematic monomineralic and polymineralic aggregates at mid-crustal conditions.This estimate was not attempted yet.This analysis is validated by comparison with experimental data on deformation of polyphase mixtures.
In this study we analyse a sample of mylonitic shear zone within the Rieserferner granodiorite (sample ID: 10-019A; sample coordinates: 46 • 55 24.8 N 12 • 07 36.2E).The structural mesoscale context of the studied sample is well illustrated by Ceccato and Pennacchioni (2018) and in particular by the map in Fig. 4 of their Supplement.

Analytical methods
Polished thin sections of granodiorite mylonite were prepared for the study of the microstructure and of the crystallographic preferred orientations (CPOs).The thin sections were made from rock chips cut parallel to the stretching lineation and perpendicular to the shear plane (XZ plane of finite strain ellipsoid).
Electron backscattered diffraction (EBSD) analysis was carried out on a JEOL 7001 FEG SEM equipped with a NordLys Max EBSD detector (AZTec acquisition software, Oxford Instruments) at the Electron Microscopy Centre of Plymouth University.EBSD patterns were acquired on rectangular grids with step sizes of 0.2, 0.3, and 0.35 µm.Working conditions during acquisition of EBSD patterns were 20 kV, 70 • sample tilt, high vacuum, and a working distance between 17 and 23 mm.A detailed description of the EBSD post-processing methods and of the image analysis are reported in Appendix A. The microstructural and CPO analysis conducted with EBSD were complemented with cathodoluminescence (CL) and microchemical analyses.
CL imaging was performed in a FEI Quanta 200 FEI equipped with Gatan monocle detector.Imaging was performed using an accelerating voltage of 20 kV, beam current of 8 nA and a working distance of 20 mm in C-coated (15 nm) thin sections used for EBSD analysis.To avoid incorrect interpretation of potential artefacts in the sample, secondary electron and backscatter electron (BSE) images were collected simultaneously with CL.
Microchemical analyses were performed with wavelengthdispersive spectroscopy (WDS) at the Electron Microprobe Laboratory at the Università degli Studi di Milano with a Jeol 8200 Super Probe; the operating conditions were 15 kV accelerating voltage, 5 nA (Kfs and Plg) beam current.The Pouchou and Pichoir (PAP) correction programme was applied to convert X-ray counts into oxide weight percentages.

Sheared myrmekite: plagioclase + quartz aggregates
Plg + Qtz aggregates (± rare Kfs and Bt) wrap around Kfs porphyroclasts and are elongate into the foliation (Fig. 1e).These aggregates make transition to, and extend into the foliation from, pristine myrmekite and are hereafter referred to as sheared myrmekite.
Qtz grains in sheared myrmekite occur either as isolated single grains at triple and quadruple junctions between Plg grains or, less commonly, as polycrystalline aggregates elongated normal to the foliation (Fig. 4a).Qtz grains within sheared myrmekite have no CPO (Fig. 4b), show little internal distortion, and show rare low-angle boundaries with scattered misorientation axis distribution (Fig. 4c).Misorientation angle distribution for correlated pairs displays higher frequency than a random-pair distribution for misorientations < 15 • and at 60 • (Fig. 4d).The uncorrelated misorientation angle distribution approaches the random-pair distribution.
Plg grains do not show any obvious CPO (Fig. 4e), and display little internal distortion and rare low-angle boundaries.The low-and high-angle misorientation axes in crystal coordinate system are almost uniformly distributed (Fig. 4f).Even though very close to random-pair distribution, correlated misorientation distribution exhibits two distinct peaks at very low angles (< 5-10 • ) and close to 180 • (Fig. 4g).Misorientations < 70 • occur with slightly higher frequency than the random-pair distribution.Albite twins and related 180 • misorientations are rarely observed inside new grains (Figs.3a-4a).In CL both myrmekitic Plg and Qtz have a grey shade similar to the surrounding non-myrmekitic Plg and Qtz (similar to Hopson and Ramseyer, 1990;Fig. 1f, h).

K-feldspar aggregates in strain shadows
In this section, EBSD data are used to describe the relationship between Kfs neoblasts and porphyroclasts.Kfs neoblasts occur in strain shadows around feldspar porphyroclasts, as well as dispersed within the sheared myrmekite (Fig. 3a, Area C in Figs.4a; S2).In strain shadows, the orientation of (100) and (010) planes and [001] direction of the neoblasts is similar to that of the porphyroclast (Fig. 4h-j).In particular, the Kfs neoblasts show a CPO for (010) planes close to the Y kinematic axis (Fig. S2d), which is similar to the orientation of (010) in the adjacent porphyroclast.Misorientation axis-angle distributions show very few scattered data without any clear clustering (Fig. 4k).The grain size of new Kfs grains dispersed within sheared myrmekite is ca.7 µm, comparable to that of the Plg in the surrounding  sheared myrmekite (Fig. 5l).The CL imaging of Kfs highlights a complex microstructure, which is different between new grains and porphyroclasts.The porphyroclasts show a homogeneous bright shade overprinted by a complex pattern of dark-grey CL shade (indicated by black arrows in Fig. 3b,  d, f).Kfs grains in sheared myrmekite and tails around porphyroclasts show a homogeneous dark-grey CL shade (indicated by black arrows in Fig. 1f-h).Kfs aggregates elongated parallel to the foliation and enveloped by sheared myrmekite are characterized by bright irregularly shaped Kfs cores (porphyroclasts) surrounded by dark-grey shaded Kfs.

Quartz layers along foliation
Monomineralic Qtz layers defining the mylonitic foliation (Figs. 2,3,4 and 5) show a variable grain size, and a shape preferred orientation (SPO) weakly inclined to the foliation consistently with the sense of shear.Dauphiné twin boundaries are widespread (red boundaries in Fig. 5a).The Qtz caxis CPO defines an asymmetric Type-II girdle inclined at high angle to the local mylonitic foliation (Fig. 5b).The pole figures of c axis and a directions show maxima roughly aligned with the Y and the X kinematic directions, respec-tively.Misorientation axis distribution for low-angle misorientation (< 10 • ) exhibits a wide maximum close to c axis and π − π directions in crystal coordinates.These misorientation axes preferentially cluster close to (but slightly offset from) the Y kinematic direction in sample coordinates (Fig. 5c).High-angle misorientation axis distributions do not show any clear systematic pattern, except for misorientations around 60 • .Misorientation angle distribution (Fig. 5d) shows two peaks at very low-angle misorientations (< 10 • ) and around 60 • for correlated misorientations.Uncorrelated misorientation angle distribution is close to the random-pair distribution.CPOs and misorientation data of coarser grains do not differ from those of finer grains.In CL images the Qtz layers display an overall homogeneous signature, with darker-grey shades close to inclusions and layer boundaries (central quartz layers in Fig. S3b).

Phase spatial distribution, grain size, and aspect ratio
The results of image analysis of EBSD phase maps indicate that pristine and sheared myrmekite have the same phase ratio with ca.18 % vol of Qtz.We have analysed the phase spatial distribution of Plg and Qtz in both pristine and sheared www.solid-earth.net/9/1399/2018/Solid Earth, 9, 1399-1419, 2018 myrmekite to define their deviation from a random distribution, either towards a clustered or an anti-clustered distribution (Heilbronner and Barrett, 2014).Phase spatial distribution analysis of a two-phase aggregate compares the cumulative lengths of phase boundaries (boundaries between grains of a different phase) and of grain boundaries (boundaries between grains of the same phase) with those expected for a random distribution of the two phases.We have considered three types of boundaries: (i) Plg-Plg grain boundaries; (ii) Qtz-Qtz grain boundaries; and (iii) Plg-Qtz phase boundaries.The results (Fig. 6) show that, in pristine and sheared myrmekite (i) the surface area fraction of Qtz ranges between 0.55-0.75 and 0.55-0.65,respectively; (ii) Qtz-Qtz grain boundaries occur with a lower probability than for a random distribution, indicative of an anti-clustered distribution; (iii) Plg-Plg grain boundaries occur with a higher probability than for a random distribution indicative of a more clustered distribution; and (iv) Plg + Qtz aggregates display an anti-clustered distribution, with Plg-Qtz phase boundaries occurring with higher probability than for random distribution of phases.The results obtained from the analysis of the area shown in Fig. 6 are consistent with the results obtained from five other areas (not presented here for the sake of brevity).
Monomineralic Qtz layers along the foliation show a variable grain size, usually ranging between 10 and 120 µm, mimicking a bimodal grain size distribution with maxima centred at 20-35 and 50-70 µm (Figs.7d and S4).The coarser grain size (> 40 µm) is observed close to the centre of Qtz layers.These grains are usually characterized by subgrains ranging in size between 20 and 35 µm.The smaller grain size (< 40 µm) commonly envelopes the coarser grains, in addition to prevailing at the boundary between monomin-

The rheology of the Rieserferner mylonites
The rheological effect of transformation of coarse Kfs to finegrained sheared myrmekite and the transition to an interconnected, weak, fine-grained microstructure (Handy, 1990) is estimated here by investigating the deformational behaviour of different mixtures of Plg and Qtz, in which deformation is accommodated either by dislocation creep or by diffusion creep.Our simplified model does not include Bt, Wmca or Bt + Plg aggregates.Based on the deformation mechanisms identified from the microstructural analysis, deformation mechanism maps have been calculated and plotted as grain size vs. differential stress and on differential stress vs. strain rate diagrams for the following three end-member compositions (Fig. 9): i. monomineralic Qtz layer deforming via both dislocation and diffusion creep (Fig. 9a); ii. sheared myrmekite, modelled as 80 % vol Plg (An 60 )+ 20 % vol Qtz deforming via both dislocation creep and GSS creep (Fig. 9b); the input grain size is 7 µm, identical for both minerals; iii. a mixture of 60 % Plg (An 100 )+ 40 % Qtz assumed as a simplified composition representative of a mica-free granitoid rock deforming only by dislocation creep (after referred to as "granitoid", Fig. 9c).
The flow law of Hirth et al. (2001) has been used to calculate the dislocation creep component in deformation mechanisms maps for Qtz: where A q is the pre-exponential factor for Qtz (MPa −n s −1 ); f h is the water fugacity; σ is the differential stress (MPa); n is the stress exponent; Q q is the activation energy (J); R is the gas constant (J K mol −1 ); and T is the temperature (K).
The contribution of diffusion creep in quartz is considered to come from pressure-solution creep and has been calculated using the flow law for thin-film pressure-solution creep of den Brok (1998): where C 2 is a shape constant; ρ f and ρ s are the fluid and solid densities (kg m −3 ), respectively; d is the grain size (µm); V is the molar volume (µm 3 mol −1 ); c is the solubility of the solid in the fluid phase (molar fraction); and D w is the diffusivity of the solid in the grain-boundary fluid film (µm 2 s −1 ).
The flow laws for poly-mineralic aggregates (e.g.sheared myrmekite and mica-free granitoid) have been calculated following the approach of Dimanov and Dresen (2005) and Platt (2015).The method allows a poly-mineralic aggregate flow law to be constructed by considering the proportional contribution of the minerals in the aggregate.The resulting flow laws for the polymineralic aggregates can be derived for both a GSS and GSI rheology and are outlined in detail in the Supplement.In our calculations only quartz and feldspar are considered as minerals of the aggregates.For quartz the flow laws used are those above (Eqs.1 and 2), while for feldspar, the flow laws of Rybacki et al. (2006) have been used to calculate the contribution of dislocation and diffusion creep, respectively: where εdisl represents the strain rate component given by dislocation creep; εdiff represents the strain rate component given by diffusion creep; A f is the pre-exponential factor for feldspar (MPa −n µm m s −1 ); d is the grain size (µm); m is the grain-size exponent; p is the confining pressure (MPa); and V act is the activation volume (m 3 mol −1 ).Flow law parameters are listed in Table 1.Details on the derivation of the deformation mechanism maps and on the calculation of the flow laws are given in the Supplement.The www.solid-earth.net/9/1399/2018/Solid Earth, 9, 1399-1419, 2018  The flow-law parameters defined for An 100 and An 60 by Rybacki and Dresen (2004) have been adopted for our calculations to simulate different compositions of "granitoid" and myrmekitic feldspars.These are "wet" flow-law parameters that have been derived experimentally from deformation of fine-grained aggregates of An 100 and An 60 containing 0.004 wt % and 0.3 wt % of water, respectively.In our calculations, all the Kfs has been considered as Plg, given the lack of flow-law parameters for Kfs (see discussion in Platt, 2015;Viegas et al., 2016).Our calculation includes the contribution of grain boundary sliding (GBS) to the bulk strain rate of the feldspar aggregate, which is considered in the flow-law parameters adopted here (see discussions in Xiao et al., 2002;Rybacki and Dresen, 2004).
The grain size vs. differential stress and differential stress vs. strain rate diagrams in Fig. 9 suggest the occurrence of different rheological behaviours that can be interpreted in terms of strain partitioning between aggregates with different "compositions".The results indicate that the three considered types of aggregates can be ranked, from the strongest to the weakest, as follows: (i) Qtz-feldspar "granitoid" agwww.solid-earth.net/9/1399/2018/Solid Earth, 9, 1399-1419, 2018 (d) Log differential stress vs. log strain rate diagram reporting the curve calculated for pure Qtz, sheared myrmekite, and ideal granitoid rock and the curves representing the rheology of a granitoid (60 % An 100 Plg+ 40 % Qtz) with a variable amount of sheared myrmekite (80 % An 60 Plg + 20 % Qtz).Maximum replacement is limited to 20 % of initial feldspar (see text for explanation).Transparent red polygon defines the stress-strain rate conditions for 15 % vol substitution of myrmekite (see text for explanation).The small figure inset shows the logarithm of the ratio between the strain rate of an ideal granitoid rock vs. that of a granitoid with increasing myrmekite substitution at the differential stress of 70 MPa.References: (1) Hirth et al. (2001).

Discussion
The following discussion is structured in two main parts: firstly, the microstructural processes controlling nucleation and the deformation mechanisms during myrmekite shearing are discussed; then, the results of rheological calculation are discussed in the light of the previously identified deformation mechanisms in order to give a quantitative estimate of the rheological effects associated with the activity of different deformation mechanisms in mono-and polymineralic domains.

Formation and shearing of myrmekite
In the following sections, EBSD data presented earlier are discussed in order to gain new insights into the myrmekite nucleation process in the light of previously published works.
The possible mechanisms involved during the transition from pristine to sheared myrmekite are then addressed, comparing the evolution of myrmekite to that of similar symplectites during shearing.Then, deformation mechanisms during A. Ceccato et al.: Myrmekite and strain weakening shearing of myrmekite are finally identified based on microstructures and EBSD data.

Crystallographic relationship between K-feldspar and myrmekitic phases
The EBSD analysis indicates that the Kfs and the overgrowing myrmekitic Plg have a similar crystallographic orientation, although with some scattering (Fig. 3b; Wirth and Voll, 1987).This suggests the occurrence of a topotactic replacive process where (100) Kfs (100) Plg , ( 010 (Stel and Breedveld, 1990;Abart et al., 2014).As observed by Abart et al. (2014), the different myrmekite Qtz vermicules have a similar crystallographic orientation.The anti-clustered phase spatial distribution of pristine myrmekite is related to the process of heterogeneous phase nucleation during myrmekite formation (Wirth and Voll, 1987).

Transition from pristine to sheared myrmekite (plagioclase + quartz aggregates)
The sheared Plg + Qtz aggregates, wrapping Kfs porphyroclasts and elongated into the foliation, resulted from shearing of pristine myrmekite.The transition from pristine to sheared myrmekite was a dynamic process and here we try to constrain the processes involved as inferred from microstructural changes.These microstructural changes included (i) randomization of Plg CPO observed in pristine myrmekite; (ii) evolution of Plg grain size distribution from heterogeneous (ranging between 3 and 50 µm) in pristine myrmekite, to homogeneous and centred at 7 µm in sheared myrmekite (Figs. 3,4 and 8a); (iii) coarsening of Qtz grains from < 3 µm thick vermicules to rounded-to-polygonal grains as large as 10 µm in sheared myrmekite (Fig. 7).The processes of grainsize evolution are probably related to the minimization of interfacial energy in the vermicular microstructure of pristine myrmekite (e.g.Odashima et al., 2007;Dégi et al., 2010).Qtz grain coarsening reflects annealing of the pristine vermicular microstructure after the reaction front moved further into the Kfs (Fig. 3a), and was probably aided by dissolutionprecipitation processes.Qtz coarsening implies simultaneous grain-size refinement of Plg, which probably involved microfracturing with the development of local micro-cracks in myrmekitic Plg.Misorientation analysis on the few low and high misorientation angle boundaries inside pristine myrmekite (inside myrmekitic Plg) shows abrupt misorientations of as much as 8 • across such boundaries, which could be interpreted as either micro-cracks or growth features con-sidering the low internal distortion of grains (Figs. 3, S1).
Microfractures could have originated from stress concentrations within the 3-D geometrically or mechanically composite structure of myrmekite (see Fig. 2 of Hopson and Ramseyer, 1990;Dell'Angelo and Tullis, 1996;Xiao et al., 2002).Therefore, the Plg grain size in the incipiently sheared aggregate may be controlled by the spacing between Qtz vermicules in pristine myrmekite.Myrmekite was then sheared along the mylonitic foliation from the contractional sites around the Kfs porphyroclast.Then, interconnected layers of sheared myrmekite developed from foliation-parallel stretching of isolated myrmekite mantling Kfs during mylonitization (similar to Boullier and Gueguen, 1975).
8.2 Deformation mechanisms in the Rieserferner mylonites

Sheared myrmekite
The microstructures of sheared myrmekite are consistent with GSS creep, including fluid-assisted GBS (Boullier and Gueguen, 1975;White, 1977;Stünitz and Fitz Gerald, 1993;Fliervoet et al., 1997;Jiang et al., 2000;Wheeler et al., 2001;Lapworth et al., 2002;Bestmann and Prior, 2003;Kilian et al., 2011;Menegon et al., 2013).Plg and Qtz of sheared myrmekite both display (i) a weak CPO; (ii) rare low-angle boundaries without systematic pattern of misorientation axis distribution; and (iii) correlated and uncorrelated misorientation angle distributions close to the theoretical random-pair distribution.All these features suggest very limited dislocation creep in both minerals (Kruse et al., 2001;Okudaira and Shigematsu, 2012;Miranda et al., 2016).In addition, sheared myrmekite show (i) fine-grained Plg and Qtz with polygonal, equant to slightly elongated shape (AR < 2); (ii) aligned grain boundaries (over the scale of several grain diameters) and common triple and quadruple junctions; and (iii) anticlustered spatial distribution of Plg and Qtz.Phase spatial distribution in deformed bimodal aggregates in mylonites is interpreted to reflect the activity of specific deformation mechanisms (Kruse and Stünitz, 1999;Menegon et al., 2013).In particular, diffusion creep in polymineralic aggregates is commonly accompanied by heterogeneous phase nucleation that promotes phase mixing and a high degree of anti-clustering in phase distribution (Kilian et al., 2011;Menegon et al., 2013).The occurrence of Qtz in triple and quadruple junctions and Qtz aggregates elongated orthogonal to the foliation in sheared myrmekite suggests creep cavitation and heterogeneous Qtz nucleation during GSS creep of Plg (Fusseis et al., 2009;Herwegh et al., 2011;Kilian et al., 2011).Heterogeneous phase nucleation in creep cavities led to the anti-clustered phase spatial distribution (Fig. 6; Hiraga et al., 2013;Menegon et al., 2015).The constant Plg grain size of sheared myrmekite may then result from the combination of initial spacing between Qtz vermicules in pristine myrmekite, diffusion creep processes, and second-phase grain-boundary pinning during shearing (Herwegh et al., 2011).GSS processes, phase mixing and second-phase grain-boundary pinning, inhibit grain growth and stabilizes grain size, hindering the efficiency of dynamic recrystallization processes and self-sustaining the activity of GSS processes.

K-feldspar tails and neoblasts
Kfs is abundant in the low-strain portions of the mylonite (Fig. 2).Kfs porphyroclasts and tails do not show any CPO or misorientation axis distribution referable to dislocation creep processes (Figs. 4i, S2;Menegon et al., 2008, and reference therein).The similar crystallographic orientation between feldspar(s) porphyroclasts and either Kfs tails or fine neoblast aggregates can be explained by invoking epitaxial nucleation and growth during dissolution-precipitation (Figs. 4, S2).Dissolution-precipitation would be consistent with the Kfs aggregate microstructure observed under CL, which probably reflect either the different chemistry or the different intra-granular strain observed between magmatic (Or 93 Ab 7 ) and synkinematic Kfs (Or 96 Ab 4 ) (Shimamoto et al., 1991;Ramseyer et al., 1992;Götze et al., 1999;Słaby et al., 2008Słaby et al., , 2014)).The modification of the inherited CPO in finegrained aggregates could be then related to the occurrence of anisotropic dissolution-precipitation processes and grain boundary sliding during myrmekite shearing (Behrmann and Mainprice, 1987;Menegon et al., 2008Menegon et al., , 2013)).

Monomineralic quartz layers
The microstructures, CL signatures, and CPO of monomineralic Qtz layers indicate deformation by dominant dislocation creep aided by sub-grain rotation (SGR) recrystallization (e.g.Fliervoet et al., 1997;Wheeler et al., 2001;Stipp et al., 2002;Bestmann and Pennacchioni, 2015).The misorientation axes distributions suggest the preferential activation of {m} a and {r − z} a slip systems (e.g.Ceccato et al., 2017, and references therein).The analysis of the grain orientation spread (GOS), to distinguish different generation of relict and/or recrystallized grains (Cross et al., 2017), suggests that there are no meaningful correlations between grain size and average grain distortion.This missing correlation may reflect a non-steady-state Qtz microstructure during a prolonged deformation history or, more likely, the development of the microstructures at different temperature conditions during pluton cooling.The bimodal grain size of recrystallized Qtz includes coarser grains that we infer developed during the relatively high-temperature bulk solid-state deformation of the host granodiorite predating the development of localized shear zones at 450 • C dominated by SGR recrystallization (Ceccato et al., 2017;Ceccato and Pennacchioni, 2018).Coarser grains in Qtz layers (grain sizes from > 40 µm) record differential stresses < 40 MPa and strain rates of 10 −13 -10 −14 s −1 as retrieved by applying the grain size paleopiezometer of Cross et al. (2017).Sub-grain and finer grains (20-35 µm in diameter) suggest that localized deformation and shearing occurred at differential stresses close to 40-70 MPa and strain rates of 10 −11 -10 −12 s −1 (Stipp and Tullis, 2003;Cross et al., 2017).

The effect of myrmekite on strain localization
In the following, the results of rheological calculations are discussed in terms of their implications on strain localization in granitoid rocks.Firstly, we will address the different rheology and strain partitioning at the microstructural scale, considering the calculated rheology for the different mono-and poly-mineralic layers.Then, the effect of increasing myrmekite substitution on the bulk rheology of a granitoid rock is addressed, comparing the rheology of the simplified granitoid rock to that of a granitoid rock with increasing myrmekite substitution.
Our results show that in the Rieserferner mylonites an effective strength contrast between mono-and poly-mineralic layers occurs at the scale of the thin section as a consequence of the different deformation mechanisms.To quantify the effective strength contrast between the modelled layer compositions, we consider two end-member conditions: constant stress and constant strain rate.Assuming that the differential stress of 40-70 MPa, estimated from the finer grain size of Qtz (from 20 to 35 µm), is representative of the bulk flow stress of the mylonite, the Qtz aggregates deforming by dislocation creep (Fig. 9a) would flow at a strain rate of 10 −11 -10 −13 s −1 , whereas sheared myrmekite deforming via diffusion creep would flow at strain rates faster than 10 −12 s −1 , depending on the actual grain size of the aggregate (red transparent area in Fig. 9b).For the grain size range of sheared myrmekite (4-7 µm), the observed strain rates are always faster than 10 −11 s −1 , and for the abovedefined differential stress range the calculated strain rate is of the order of 10 −9 s −1 (intersection between red transparent area and black box in Fig. 9b).Therefore, assuming constant differential stress conditions, a strain-rate partitioning of 2-4 orders of magnitude is expected between monomineralic Qtz and sheared myrmekite (similarly to Behrmann and Mainprice, 1987).Assuming constant strain rate conditions of 10 −11 -10 −12 s −1 , the differential stress calculated for sheared myrmekite deforming via diffusion creep is < 45 MPa.Under the constant strain rate assumption, the strength contrast between monomineralic Qtz and sheared myrmekite is not quantifiable; however, the sheared myrmekite are always weaker than monomineralic Qtz deforming via dislocation creep.Strain rates of the order of 10 −11 -10 −13 s −1 would require grain sizes in the range of 10-100 µm in the sheared myrmekite deforming by diffusion creep only (grey shaded areas in Fig. 9b).
Figure 9d shows the different curves describing the rheological behaviour of a simplified granitoid rock where Kfs is progressively replaced, up to 20 % vol, by sheared www.solid-earth.net/9/1399/2018/Solid Earth, 9, 1399-1419, 2018 myrmekite.The flow behaviour of the derived granitoid mylonite is represented by the grey curves, and is linear viscous for most of the investigated conditions.The complete consumption of Kfs results in 3-4 orders of magnitude increase in strain rate at differential stress conditions of 70 MPa (inset of Fig. 9d), consistent with experimental observations (Xiao et al., 2002).A similar increase in strain rate is already observed for a 5 % vol of sheared myrmekite in the total rock volume.These results can be compared to the different degree of myrmekite substitution observed along the strain gradient in the shear zone and also could explain the progressive increase in strain toward the ultramylonite with increasing myrmekite substitution (Fig. 2), suggesting positive feedback between strain-induced myrmekite formation and strain accommodation.Dissolution-precipitation creep of Kfs and associated GSS creep in Kfs + Plg + Qtz aggregates have been already described by Behrmann and Mainprice (1987) as an efficient strain accommodation and weakening mechanism in Qtz-feldspar mylonites.In the Rieserferner mylonites, GSS creep of Kfs seems to be dominant in protomylonite, but its role decreases with increasing myrmekite substitution (Fig. 2).The positive correlation between accommodated strain and amount of sheared myrmekite suggests that GSS creep processes in Kfs are not capable of accommodating strain at rates comparable to those produced by GSS creep in sheared myrmekite.
The effect of myrmekite development in rheological weakening might be overestimated by our calculation for two main reasons: (i) other weakening mechanisms, which are not considered in our simplified model of granitoid (such as feldspar GSS creep, Bt deformation), may have concurred with deformation; and (ii) at low strain, myrmekite aggregates were initially non-interconnected pockets (e.g.Handy, 1994).Strain weakening associated with myrmekite is inferred to become relevant as, with increasing strain and volume fraction of sheared myrmekite, the initially isolated myrmekite are sheared and coalesced into an interconnected network.In the Rieserferner sheared granodiorites, an interconnected framework of sheared myrmekite is established in the presence of 5 % vol to 7 % vol of myrmekite and is well developed at 10 % vol-15 % vol (Fig. 2).Therefore, mylonites containing up to 15 % vol of sheared myrmekite ideally underwent deformation at strain rates of 10 −10 -10 −11 s −1 and at differential stresses in the range between 10 and 70 MPa (transparent red area bracketed by the curves calculated for 10 % vol and 20% vol of myrmekite substitution in Fig. 9d).These mylonites were synkinematic to mylonitic Qtz veins described in Ceccato et al. (2017), for which Qtz paleopiezometry retrieved comparable strain rates of 10 −11 s −1 for 117 MPa differential stress.

Conclusions
Metamorphic reactions contributed importantly to strain weakening within the Rieserferner granitoid mylonites.A primary grain-size reduction mechanism was related to the development of myrmekite evolving, with increasing strain, to weak aggregates of Qtz and Plg.Topotactic replacement has been inferred from the coincidence between myrmekitic Plg and parent Kfs grain crystal lattices in pristine myrmekite.Transition from pristine myrmekite to finegrained sheared myrmekite involved microfracturing, annealing, and shearing of the resulting granoblastic aggregate.Sheared myrmekite consists of fine-grained Plg + Qtz aggregates (7 and 4 µm in grain size, respectively) that show anti-clustered spatial distribution and well-defined SPO; Qtz usually occurs at triple and quadruple junctions between Plg grains.Both Plg and Qtz show weak CPOs and almost uniform misorientation angle distributions.The microstructures of sheared myrmekite suggest that different deformation mechanisms occurred in Plg and Qtz: Plg mainly deformed by GSS creep, whereas dissolution-precipitation and nucleation processes were dominant in Qtz.Myrmekite formation also promoted phase mixing, as the pristine myrmekite microstructure predisposed the development of an "anticlustered" spatial distribution of phases in the recrystallized aggregate.Strong grain-size reduction and the nucleation of Plg + Qtz polymineralic aggregates led to a switch in the dominant deformation mechanisms, activating GSS creep processes and triggered phase mixing.GSS processes and phase mixing inhibited grain growth and stabilized grain size, hindering the efficiency of dynamic recrystallization by dislocation creep processes and self-sustaining the activity of GSS processes.Therefore, the formation of myrmekite led to the activation of self-sustaining weakening processes.Results of rheological calculations show that, at the conditions of Rieserferner mylonitization, sheared myrmekite are several orders of magnitude weaker than both pure Qtz layers and ideal granitoid rock deforming via dislocation creep.Strain-rate partitioning is therefore expected to occur between sheared myrmekite and monomineralic Qtz layers, and the occurrence of ca. 5 % vol of myrmekite could lead to an increase of 3-4 orders of magnitude in strain rate.However, the effective role of myrmekite in rock weakening depends on the evolution of the rock microstructure.Effective weakening requires interconnection of sheared myrmekite layers, which occurs after the development of 10 % vol-15 % vol of myrmekite.This work highlights the importance of metamorphic reactions as grain-size reduction mechanisms in feldspar, and their role in localization of ductile deformation via the activation of GSS creep.The microstructural results and the rheological calculation presented here will be useful for further development of detailed rheological models of feldspar-rich rocks at mid-crustal conditions.
A. Ceccato et al.: Myrmekite and strain weakening Appendix A: Methods

A1 EBSD sample preparation and data processing
The thin section was SYTON-polished for ca. 3 h and carbon coated.All data have been processed and analysed using CHANNEL5 software of HKL Technology, Oxford Instruments.Noise reduction was applied following Bestmann and Prior (2003).Local mis-indexing between Plg and Kfs was resolved by nullifying the subset of selected grains with area < 1 µm 2 in each map.Dauphiné twins smaller than 0.5 µm have been interpreted as an error from mis-indexing and were replaced by the average orientation of the neighbouring pixels.The indexed phases and relative symmetry group used for the indexing are quartz -trigonal -3m; plagioclase (anorthite) -triclinic -1; and orthoclase -monoclinic 2/m.Critical misorientation for the distinction between low-and highangle boundaries have been chosen at 10 • .Qtz grain boundaries with 60 • ± 5 • of misorientation were disregarded from the grain detection procedure to avoid any contribution from Dauphiné twinning.Plg grain boundaries with 180 • ± 5 • of misorientation around [010] were disregarded from the grain detection procedure to avoid any contribution from Albite twinning.The pole figures (one point per grain, where not differently specified) are plotted as equal area, lower hemisphere projections oriented with the general shear zone kinematics reference system (X = stretching lineation; Z = pole to general shear plane/vein boundary); whereas the misorientation axis distributions in sample coordinates are plotted as equal area, upper hemisphere projections.The inverse pole figures for misorientation axis distribution in crystal coordinates are upper hemisphere projections.Contoured projections have constant contouring parameters (halfwidth: 10 • ).Contouring lines are given only for the 0.5-10 m.u.d.(multiple of uniform distributions) range.

A2 Grain size analysis
Grain sizes were obtained from the grain detection routine of the HKL Channel5 Tango software.The grain size was calculated as diameter of the circle with an equivalent area.The minimum cut-off area was set to 1 µm 2 , which means that only grains of a size ≥ 4 or ≥ 9 pixels (depending on the map acquisition step size) were considered.Grain size data were represented as area-weighted distributions by plotting frequency against the square-root grain-size-equivalent grain diameters (as in Herwegh and Berger, 2004;Berger et al., 2011).The grain size distribution approaches a Gaussian distribution when plotted in this way, allowing a good estimate of the mean grain size.The geometric mean grain size (red thick line in grain size distribution diagrams) was obtained graphically as the maximum frequency grain size of the distribution curve.

A3 Image analysis
Image analysis of grain shape was performed on both scanning electron microscope backscattered electron (SEM-BSE) images and phase maps obtained from EBSD.Quantification of phase amount (% vol) was performed through segmentation of SEM-BSE images of a whole thin section collected at the Electron Microscopy Centre of the University of Plymouth.Image processing and thresholding was done with the ImageJ software, and further processing together with manual correction were applied to improve data quality and to ensure the correspondence of greyscale ranges with specific mineral phases.Grain boundary images and phase distribution images were obtained directly from EBSD phase maps and grain boundary maps elaborated by Channel5 (HKL Technology).Before the analysis with ImageJ software, images were manually corrected in order to exclude misindexing and non-indexed orientation pixels.Grain boundaries and phase amount have been quantified by pixel counting.
Author contributions.AC, LM and GP developed the initial idea of the study and performed the initial exploratory SEM study.GP collected the samples of Rieserferner mylonites.LM acquired EBSD data.AC performed EBSD data processing and analysis, and the rheological calculations.LFGM performed cathodoluminescence analysis.AC prepared the figures and the manuscript with contributions from all the co-authors.
Competing interests.The authors declare that they have no conflict of interest.

Figure 1 .
Figure 1.Microstructures of Rieserferner granodiorite mylonites.(a) Micrograph (crossed polarizers) showing the alternating layers of recrystallized Qtz, recrystallized Bt + Plg + Qtz, and Plg + Qtz.White arrows indicate layers of recrystallized Qtz (upper) and Bt (lower).(b) SEM-BSE image of the area shown in (a).(c) SEM-BSE image of a pristine myrmekite (Myrm) replacing Kfs.Plg + Ttn symplectites (Symp) are indicated by white arrows.(d) SEM-BSE image of the Kfs + Bt tails in strain shadows between two Plg porphyroclasts.Kfs in the strain shadows is in turn replaced by myrmekite (white arrows).(e) SEM-BSE image of a Kfs porphyroclast and of sheared myrmekite.Pristine myrmekite developed on Kfs boundaries parallel to the mylonitic foliation are sheared to form Plg + Qtz aggregates (sheared myrmekite).The white polygon encloses Kfs neoblasts in strain shadows and sheared myrmekite.Plg + Ttn symplectites (Symp) are indicated by white arrows.(f) CL image of (e).Black arrows indicate the Kfs in strain shadows and porphyroclast tails showing darkgrey homogeneous CL shades.Note the alteration of the CL signal in Qtz after the EBSD scan (area delimited by white dashed line).(g) Kfs and sheared myrmekite aggregate (particular of the EBSD map of Fig. 3).(h) CL image of (g).

Figure 2 .
Figure 2. Phase distribution and abundance across a strain gradient in a granodiorite mylonite.(a) Mosaic of SEM-BSE images with the Kfs and the myrmekite + sheared myrmekite coloured in red and pale blue, respectively, and Qtz coloured in black.The yellow rectangles indicate the location of the EBSD maps of Figs. 3, 4, 5, and S2.(b) Bar diagram showing the volume amount of Kfs (red bars) and myrmekite (pale blue bars) across the microstructure: PM represents protomylonite; M represents mylonite; and UM represents ultramylonite.

Figure 3 .
Figure 3. EBSD map and crystallographic orientation data of incipient myrmekite and parent Kfs.(a) EBSD-derived phase map.The area delimited by dashed polygons represents pristine myrmekite.Grain boundaries are colour-coded according to the maximum angle of misorientation observed across them.Pole figures for (b) Kfs grains on which pristine myrmekite nucleated; (c) Plg and (d) Qtz in pristine myrmekite.(e) Enlargement of EBSD map in (a) showing the low-angle (and twin, red) boundaries in plagioclase abutting against quartz vermicules (indicated by white arrows).

Figure 4 .
Figure 4. EBSD map and crystallographic orientation data of pristine and sheared myrmekite of Fig. 1e.(a) EBSD phase map including areas (A, B, C, D) selected for grain size analysis and phase distribution analysis.(b) Pole figures for Qtz from the sheared myrmekite of Area B. Upper row: scattered data.Lower row: contoured data.(c) Misorientation axis distributions for Qtz in sample (upper row) and crystal (lower row) coordinate system.(d) Misorientation angle distribution for Qtz.(e) Pole figures for Plg from sheared myrmekite of Area B. Upper row: scattered data.Lower row: contoured data.In this case, the [100] Plg pole figure is reported in upper hemisphere, where the maximum has been observed.(f) Misorientation axis distributions for Plg in sample (upper row) and crystal (lower row) coordinate system.(g) Misorientation angle distribution for Plg.(h) Pole figures of the crystallographic orientation of Kfs porphyroclasts included in Areas C and E and respective tails.(i) Misorientation axis distributions in sample (upper row) and crystal (lower row) coordinate system for porphyroclasts and tails.(j) Pole figures of the crystallographic orientation of Kfs porphyroclast A and Kfs neoblasts in the strain shadow (Area D). (k) Misorientation axis distributions in sample (upper row) and crystal (lower row) coordinate system for Kfs neoblasts.(l) Grain size distribution for the Kfs neoblasts.

Figure 5 .
Figure 5. EBSD orientation data and mapping for pure Qtz layers.(a) Orientation map colour coded according to the inverse pole figure for Y direction reported in the lower right corner.(b) Area-weighted grain-size distribution for pure Qtz layer.(b) Pole figures for Qtz [c], a and {r} crystallographic elements.(c) Misorientation axis distributions in sample (upper row) and crystal (lower row) coordinate system.(d) Misorientation angle distribution for Qtz.

Figure 6 .
Figure 6.Image analysis of the phase spatial distribution in myrmekite.The diagram reports phase-and grain-boundary fractions in pristine and sheared myrmekite.Continuous curves represent the theoretical probability of phase-and grain-boundary fraction as a function of Qtz content expected for a random distribution in a two-phase aggregate.The small maps on the left-hand side report one of the analysed areas (Area C. Fig. 4), showing from the top to the bottom the phase map, the Plg grain boundaries, the Qtz grain boundaries, and the Plg-Qtz phase boundaries.

Figure 7 .
Figure 7. Area-weighted grain size distributions and SPO for Qtz.(a) Grain-size distribution for Qtz in incipient myrmekite A in Fig. 4a.(b) Grain size distribution for Qtz in sheared myrmekite B in Fig. 4a.(c) Grain-size distribution for Qtz in sheared myrmekite C in Fig. 4a.(d) Grain-size distribution for Qtz in monomineralic layer in Fig. 4a.(e) Relative frequency distribution of grain aspect ratio for Qtz.(f) Rose diagram showing the orientation of major axis of Qtz grains, defining a weak SPO.The distribution curve (blue line in grain size distribution diagrams) was obtained by interpolating distribution data with a sixth-degree polynomial equation in MS Office Excel.Relative frequencies are normalized to 1.

Figure 8 .
Figure 8. Area-weighted grain-size distributions and SPO for Plg.(a) Grain-size distribution for Plg in myrmekite of Fig. 3. (b) Grain-size distribution for Plg in incipient myrmekite A in Fig. 4a.(c) Grain-size distribution for Plg in sheared myrmekite B in Fig. 4a.(d) Relative frequency distribution of grain aspect ratio for Plg.(e) Rose diagram showing the orientation of major axis of Plg grains, defining a weak SPO.The distribution curve (blue line in grain-size distribution diagrams) was obtained by interpolating distribution data with a sixth-degree polynomial equation in MS Office Excel.Relative frequencies are normalized to 1.

Figure 9 .
Figure 9. Diagrams derived from the calculation of the rheological model explained in the text.Grain size vs. differential stress map with contoured strain rate curves calculated for (a) Qtz and (b) 80 % Plg (An 60 )+ 20% Qtz aggregates.(a) The piezometric curve from Stipp and Tullis (2003, black curve) and Cross et al. (2017, red curve) are reported.Red and black stars mark the differential stress/strain-rate conditions defined by the grain size observed in pure Qtz layers: (a) 35 µm; (b) 20 µm; (c) 10 µm(Ceccato et al., 2017).(b) The black dashed line represents the boundary between dislocation and diffusion creep dominated conditions.The black rectangle represents the grain size range (4-7 µm) observed in the sheared myrmekite.The grey semi-transparent polygon defines the field of possible grain-size and differential stress conditions for iso-strain-rate conditions defined from piezometric relations.(c) Log differential stress vs. log strain rate diagram reporting the curve calculated for pure Qtz with different grain sizes (4, 10, 20, 100 µm), sheared myrmekite, ideal granitoid rock and the curves representing the rheology of pure feldspar aggregates.For comparison, one of the curve obtained from experimental data ofXiao et al. (2002)  is reported (black dashed curve).Grey field represents the uncertainties in the experimentally defined rheological curve.(d) Log differential stress vs. log strain rate diagram reporting the curve calculated for pure Qtz, sheared myrmekite, and ideal granitoid rock and the curves representing the rheology of a granitoid (60 % An 100 Plg+ 40 % Qtz) with a variable amount of sheared myrmekite (80 % An 60 Plg + 20 % Qtz).Maximum replacement is limited to 20 % of initial feldspar (see text for explanation).Transparent red polygon defines the stress-strain rate conditions for 15 % vol substitution of myrmekite (see text for explanation).The small figure inset shows the logarithm of the ratio between the strain rate of an ideal granitoid rock vs. that of a granitoid with increasing myrmekite substitution at the differential stress of 70 MPa.

Table 1 .
Parameters adopted in the rheological calculations.(a) List of the general parameters adopted in the rheological calculations.(b) Values of flow-law parameters adopted in the rheological calculations according to mineral phase and deformation mechanism.
) Kfs (010) Plg , and (001) Kfs (001) Plg .The scatter in crystallographic orientation between Kfs and myrmekitic Plg is interpreted to result from deformation during and after myrmekite formation (see Sect. 8.1.2).The crystallographic orientation of myrmekitic Plg and Qtz was not controlled by neighbour Plg or Qtz grains previously in contact with the Kfs, unlike what is reported by other authors