Ichiko Shimizu

Kinetics of pressure solution creep in quartz: theoretical considerations

Abstract The kinetics of pressure solution creep are formulated using chemical potentials generalized to nonhydrostatic states. Solving a coupling equation of diffusion and reaction on a spherical quartz grain with diameter d and grain boundary width w, the flow law of pressure solution creep is derived. As extreme cases, the flow law becomes: ϵ = (αν 2 SiO 2 KDw)(ν H 2 O RTd 3 ) −1 σ for the diffusion-controlled case and becomes: ϵ = (βν 2 SiO 2 κ + )(ν H 2 O RTd) −1 σ for the reaction-controlled case, where ϵ is strain rate, σ is deviatoric stress, ν is the molar volume, D is the diffusion coefficient through a wet grain boundary, K is the equilibrium constant, κ+ is the rate constant of dissolution, R is the gas constant, T is temperature, and α and β are shape factors. Using the reaction constants determined by Rimstidt and Barnes (1980) and the grain boundary diffusion coefficients estimated by Nakashima (1995), the strain rate of pressure solution creep in metamorphic conditions for quartzose rocks is estimated as 10−9∼13, 10−8∼11, and 10−7∼11 s−1 at 150, 250, and 350°C, respectively. These values, compared with the duration of regional metamorphism, suggest rapid pressure solution and dewatering in subduction zones followed by fluid-absent metamorphism.

Stress and temperature dependence of recrystallized grain size: A subgrain misorientation model

The steady-state grain size of Earth materials undergoing solid state flow is estimated based on a nucleation-and-growth model of dynamic recrystallization. Assuming a nucleation mechanism of subgrain rotation, the mean diameter d of recrystallized grains is obtained as d/b = A(σ/µ)−p exp[-((Qgb - Qv)/mkT)], where b is the length of the Burgers vector, σ is differential stress, µ is the shear modulus, Qgb is the activation energy for the jump of an atom across the grain boundary, Qv is that for self-diffusion in the grain volume, k is the Boltzmann constant, T is temperature, A is a constant, p = 1.25 and m = 4 for intracrystalline nucleation, and p = 1.33 and m = 3 for grain-boundary nucleation. The exponent p = 1.25 ∼ 1.33 agrees well with available data for high- temperature dislocation creep of rock-forming minerals. A weak negative dependence of grain size on temperature is expected from this theory.

nonhydrostatic and nonequilibrium thermodynamics of deformable materials

The nonequilibrium thermodynamics of polycrystalline solids under nonhydrostatic stress are formulated, based on the conservation laws of classical dynamics and the second law of thermodynamics. For a closed system, the thermodynamic potential function corresponding to the Gibbs free energy, in classical thermodynamics, is defined as a Legendre transformation of the internal energy of the system, whereas it cannot be defined in an open system under stress. Chemical potential is defined at source points where addition of matter is possible. Chemical potential for grain growth (μ+) under nonhydrostatic stress varies with the orientation of grain surfaces and with direction of grain growth. Pressure solution, diffusion creep, and anisotropic phase equilibrium under stress are explained by the orientation dependence of μ+.

A stochastic model of grain size distribution during dynamic recrystallization

Abstract The grain size distribution (GSD) during dynamic recrystallization was investigated by a simple nucleation-and-growth model. The steady-state GSD is represented by a sum of a series of lognormal distributions, and is characterized by an asymmetric bell-shaped curve in a logarithmic frequency diagram. The distributions of one- and two-dimensional grain size (i.e. length of linear intercept and diameter of intersection area, respectively) are similar to those of the lognormal distribution. The average grain size varies with the ratio of nucleation rate and growth rate, but is hardly affected by the size of nuclei.

The non-equilibrium thermodynamics of intracrystalline diffusion under non-hydrostatic stress

Abstract Intracrystalline diffusion induced by non-hydrostatic stress is formulated on the basis of the non-equilibrium thermodynamics of deformable solids. In the case of ideal solution, the chemical potential of the diffusing component a is expressed as where v is the molar volume, σ ij is the stress tensor, f is the molar Helmholtz free energy, δ ij is the Kronecker delta, e ij represents the strain tensor of the crystal lattice, and the superscript a denotes the partial molar values of component a. The driving force of intracrystaline diffusion of component a is approximately -v(∂e ij /∂Ca)∇σ ij . The lattice diffusion creep is reinterpreted in terms of .

Theoretical derivation of flow laws for quartz dislocation creep: Comparisons with experimental creep data and extrapolation to natural conditions using water fugacity corrections

We theoretically derived flow laws for quartz dislocation creep using climb-controlled dislocation creep models and compared them with available laboratory data for quartz plastic deformation. We assumed volume diffusion of oxygen-bearing species along different crystallographic axes (//c, ⊥R, and ⊥c) of α-quartz and β-quartz, and pipe diffusion of H 2 O, to be the elementary processes of dislocation climb. The relationships between differential stress (σ) and strain rate (_ e) are written as _ e∝σ 3 D v and _ e∝σ 5 D p for cases controlled by volume and pipe diffusion, respectively, where D v and D p are coefficients of diffusion for volume and pipe diffusion. In previous experimental work, there were up to ~1.5 orders of magnitude difference in the water fugacity values in experiments that used either gas-pressure-medium or solid-pressure-medium deformation apparatus. Therefore, in both the theories and flow laws, we included water fugacity effects as modified preexponential factors and water fugacity terms. Previous experimental data were obtained mainly in the β-quartz field and are highly consistent with the volume-diffusion-controlled dislocation creep models of β-quartz involving the water fugacity term. The theory also predicts significant effects for the transition of α-β quartz under crustal conditions. Under experimental pressure and temperature conditions, the flow stress of pipe-diffusion-controlled dislocation creep is higher than that for volume-diffusion-controlled creep. Extrapolation of the flow laws to natural conditions indicates that the contributions of pipe diffusion may dominate over volume diffusion under low-temperature conditions of the middle crust around the brittle-plastic transition zone.

Rheological profile across the NE Japan interplate megathrust in the source region of the 2011 Mw9.0 Tohoku-oki earthquake

A strength profile across the NE Japan interplate megathrust was constructed in the source region of the 2011 Tohoku-oki earthquake (Mw9.0) using friction, fracturing, and ductile flow data of the oceanic crustal materials obtained from laboratory experiments. The depth-dependent changes in pressure, temperature, and pore fluid pressure were incorporated into a model. The large tsunamigenic slips during the M9 event can be explained by a large gradient in fault strength on the up-dip side of the M9 hypocenter, which was located 17 to 18 km beneath sea level. A large stress drop (approximately 80 MPa) induced by the collapse of a subducted seamount possibly triggered the M9 earthquake. In the deep (>35 km) part of the thrust fault, where M7-class Miyagi-oki earthquakes have repeatedly occurred, plastic deformation occurs in siliceous rocks but not in gabbroic rocks. Thus, the asperity associated with the M7-class earthquakes was most likely a gabbroic body, such as a broken seamount, surrounded by siliceous sedimentary rocks. The conditionally stable nature of the surrounding region can be explained by the frictional behavior of wet quartz in the brittle-ductile transition zone. In contrast to the deep M7-class asperity, the M9 asperity (i.e., a region that was strongly coupled before the M9 Tohoku-oki earthquake) extended to a large area of the plate interface because shear strength is relatively insensitive to lithological variation at intermediate depths. However, the along-arc extension of the M9 asperity was constrained by fluid-rich regions on the plate interface.

Steady-State Grain Size in Dynamic Recrystallization of Minerals

where σ is the flow stress, μ is the shear modulus, b is the length of the Burgers vector, and K is a non-dimensional constant. The grain size exponent p ranges between 1 and 1.5 for most materials. Empirically determined σ–d relations of minerals have been used to estimate the stress states in the Earth’s interior. However, detailed studies of a Mg alloy (De Bresser et al., 1998) and NaCl (Ter Heege et al., 2005) revealed that K has a weak dependence on temperature. Derby & Ashby (1987) modeled the DRX processes of metals and predicted the temperature dependence of the recrystallized grain size, but they failed to account for the observed range of exponent p (Derby, 1992; Shimizu, 2011).

Fossilized Melts in Mantle Wedge Peridotites

The shallow oxidized asthenosphere may contain a small fraction of potassic silicate melts that are enriched in incompatible trace elements and volatiles. Here, to determine the chemical composition of such melt, we analysed fossilized melt inclusions, preserved as multiphase solid inclusions, from an orogenic garnet peridotite in the Bohemian Massif. Garnet-poor (2 vol.%) peridotite preserves inclusions of carbonated potassic silicate melt within Zn-poor chromite (<400 ppm) in the clinopyroxene-free harzburgite assemblage that equilibrated within the hot mantle wedge (Stage 1, > 1180 °C at 3 GPa). The carbonated potassic silicate melt, which has a major element oxide chemical composition of K2O = 5.2 wt.%, CaO = 17 wt.%, MgO = 18 wt.%, CO2 = 22 wt.%, and SiO2 = 20 wt.%, contains extremely high concentrations of large ion lithophile elements, similar to kimberlite melts. Peridotites cooled down to ≅800 °C during Stage 2, resulted in the growth of garnet relatively poor in pyrope content, molar Mg/(Mg + Fe + Ca + Mn), (ca. 67 mol.%). This garnet displays a sinusoidal REE pattern that formed in equilibrium with carbonatitic fluid. Subsequently, subduction of the peridotite resulted in the formation of garnet with a slightly higher pyrope content (70 mol.%) during the Variscan subduction Stage 3 (950 °C, 2.9 GPa). These data suggest the following scenario for the generation of melt in the mantle wedge. Primarily, infiltration of sediment-derived potassic carbonatite melt into the deep mantle wedge resulted in the growth of phlogopite and carbonate/diamond. Formation of volatile-bearing minerals lowered the density and strength of the peridotite. Finally, phlogopite-bearing carbonated peridotite rose as diapirs in the mantle wedge to form carbonated potassic silicate melts at the base of the overriding lithosphere.

Fractal Nature of the Band-Thickness in the Archean Banded Iron Formation in the Yellowknife Greenstone Belt, Northwest Territories, Canada

Banded iron formations (BIFs) are chemically precipitated deposits on the Precambrian sea floor and are characterised by alternations of repeat Fe-rich and Si-rich layers [1]. Temporal varia‐ tions in the volumes of BIFs are considered to be related to early evolution of the atmosphere, oceans, life and the Earth’s interior [2, 3]. In general, BIFs contain various scales of banding. Bands with a thickness of several tens of meters to meters, a thickness of centimetres and a thickness of submillimetre to millimetres are named macrobands, mesobands and microbands, respective‐ ly [4]. Some depositions are related to periodic phenomena, such as annual cycles [4], tidal and solar cycles [5–7], and Milankovitch cycles [8, 9] in the Precambrian. On the other hand, quanti‐ tative analysis of the banding is limited to Paleoproterozoic Hamersley (Superior-type) BIFs, although BIFs occur within an age range from 3.8 Ga to about 0.7 Ga [10]. Therefore, it is necessa‐ ry to investigate different BIFs, in terms of both their age and type, clarified by size and litholog‐ ical facies (i.e., Superiorand Algoma-types) to understand the nature of their banded structures.