Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

This work is based upon a coupled, lattice-based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport; e.g. diffusional creep and electromigration. Here we discuss an enhancement of this formulation to account for migrating grain boundaries. The level set method is used to model grain-boundary migration in an Eulerian framework where a grain boundary is represented as the zero level set of an evolving higher-dimensional function. This approach can easily be generalized to model other problems involving migrating interfaces; e.g. void evolution and free-surface morphology evolution. The level-set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques. This simplified level-set formulation makes use of velocity extension and field re-initialization techniques. In addition, a least-squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level-set field without resorting to higher-order interpolation. A notable feature is that the coupling between mass transport, mechanics and grain-boundary migration is fully accounted for. The complexities associated with this coupling are highlighted and the operator-split algorithm used to solve the coupled equations is described.Comment: 28 pages, 9 figures, LaTeX; Accepted for publication in Computer Methods in Applied Mechanics and Engineering. [Style and formatting modifications made, references added.

Irreversible thermodynamics of creep in crystalline solids

We develop an irreversible thermodynamics framework for the description of creep deformation in crystalline solids by mechanisms that involve vacancy diffusion and lattice site generation and annihilation. The material undergoing the creep deformation is treated as a non-hydrostatically stressed multi-component solid medium with non-conserved lattice sites and inhomogeneities handled by employing gradient thermodynamics. Phase fields describe microstructure evolution which gives rise to redistribution of vacancy sinks and sources in the material during the creep process. We derive a general expression for the entropy production rate and use it to identify of the relevant fluxes and driving forces and to formulate phenomenological relations among them taking into account symmetry properties of the material. As a simple application, we analyze a one-dimensional model of a bicrystal in which the grain boundary acts as a sink and source of vacancies. The kinetic equations of the model describe a creep deformation process accompanied by grain boundary migration and relative rigid translations of the grains. They also demonstrate the effect of grain boundary migration induced by a vacancy concentration gradient across the boundary

River-bed armoring as a granular segregation phenomenon

Gravel-river beds typically have an "armored" layer of coarse grains on the surface, which acts to protect finer particles underneath from erosion. River bed-load transport is a kind of dense granular flow, and such flows are known to vertically segregate grains. The contribution of granular physics to river-bed armoring, however, has not been investigated. Here we examine these connections in a laboratory river with bimodal sediment size, by tracking the motion of particles from the surface to deep inside the bed, and find that armor develops by two distinct mechanisms. Bed-load transport in the near-surface layer drives rapid segregation, with a vertical advection rate proportional to the granular shear rate. Creeping grains beneath the bed-load layer give rise to slow but persistent segregation, which is diffusion dominated and insensitive to shear rate. We verify these findings with a continuum phenomenological model and discrete element method simulations. Our results suggest that river beds armor by granular segregation from below --- rather than fluid-driven sorting from above --- while also providing new insights on the mechanics of segregation that are relevant to a wide range of granular flows

Partially fluidized shear granular flows: Continuum theory and MD simulations

The continuum theory of partially fluidized shear granular flows is tested and calibrated using two dimensional soft particle molecular dynamics simulations. The theory is based on the relaxational dynamics of the order parameter that describes the transition between static and flowing regimes of granular material. We define the order parameter as a fraction of static contacts among all contacts between particles. We also propose and verify by direct simulations the constitutive relation based on the splitting of the shear stress tensor into a``fluid part'' proportional to the strain rate tensor, and a remaining ``solid part''. The ratio of these two parts is a function of the order parameter. The rheology of the fluid component agrees well with the kinetic theory of granular fluids even in the dense regime. Based on the hysteretic bifurcation diagram for a thin shear granular layer obtained in simulations, we construct the ``free energy'' for the order parameter. The theory calibrated using numerical experiments with the thin granular layer is applied to the surface-driven stationary two dimensional granular flows in a thick granular layer under gravity.Comment: 20 pages, 19 figures, submitted to Phys. Rev.

Protostellar Disk Evolution Over Million-Year Timescales with a Prescription for Magnetized Turbulence

Magnetorotational instability (MRI) is the most promising mechanism behind accretion in low-mass protostellar disks. Here we present the first analysis of the global structure and evolution of non-ideal MRI-driven T-Tauri disks on million-year timescales. We accomplish this in a 1+1D simulation by calculating magnetic diffusivities and utilizing turbulence activity criteria to determine thermal structure and accretion rate without resorting to a 3-D magnetohydrodynamical (MHD) simulation. Our major findings are as follows. First, even for modest surface densities of just a few times the minimum-mass solar nebula, the dead zone encompasses the giant planet-forming region, preserving any compositional gradients. Second, the surface density of the active layer is nearly constant in time at roughly 10 g/cm2, which we use to derive a simple prescription for viscous heating in MRI-active disks for those who wish to avoid detailed MHD computations. Furthermore, unlike a standard disk with constant-alpha viscosity, the disk midplane does not cool off over time, though the surface cools as the star evolves along the Hayashi track. The ice line is firmly in the terrestrial planet-forming region throughout disk evolution and can move either inward or outward with time, depending on whether pileups form near the star. Finally, steady-state mass transport is a poor description of flow through an MRI-active disk. We caution that MRI activity is sensitive to many parameters, including stellar X-ray flux, grain size, gas/small grain mass ratio and magnetic field strength, and we have not performed an exhaustive parameter study here.Comment: Accepted for publication in Astrophysical Journal. 19 pages, including 8 figure

Wind-driven Accretion in Protoplanetary Disks. I: Suppression of the Magnetorotational Instability and Launching of the Magnetocentrifugal Wind

We perform local, vertically stratified shearing-box MHD simulations of protoplanetary disks (PPDs) at a fiducial radius of 1 AU that take into account the effects of both Ohmic resistivity and ambipolar diffusion (AD). The magnetic diffusion coefficients are evaluated self-consistently from a look-up table based on equilibrium chemistry. We first show that the inclusion of AD dramatically changes the conventional picture of layered accretion. Without net vertical magnetic field, the system evolves into a toroidal field dominated configuration with extremely weak turbulence in the far-UV ionization layer that is far too inefficient to drive rapid accretion. In the presence of a weak net vertical field (plasma beta~10^5 at midplane), we find that the MRI is completely suppressed, resulting in a fully laminar flow throughout the vertical extent of the disk. A strong magnetocentrifugal wind is launched that efficiently carries away disk angular momentum and easily accounts for the observed accretion rate in PPDs. Moreover, under a physical disk wind geometry, all the accretion flow proceeds through a strong current layer with thickness of ~0.3H that is offset from disk midplane with radial velocity of up to 0.4 times the sound speed. Both Ohmic resistivity and AD are essential for the suppression of the MRI and wind launching. The efficiency of wind transport increases with increasing net vertical magnetic flux and the penetration depth of the FUV ionization. Our laminar wind solution has important implications on planet formation and global evolution of PPDs.Comment: 23 pages, 13 figures, accepted to Ap

Magmatic focusing to mid-ocean ridges: the role of grain size variability and non-Newtonian viscosity

Melting beneath mid-ocean ridges occurs over a region that is much broader than the zone of magmatic emplacement to form the oceanic crust. Magma is focused into this zone by lateral transport. This focusing has typically been explained by dynamic pressure gradients associated with corner flow, or by a sub-lithospheric channel sloping upward toward the ridge axis. Here we discuss a novel mechanism for magmatic focusing: lateral transport driven by gradients in compaction pressure within the asthenosphere. These gradients arise from the co-variation of melting rate and compaction viscosity. The compaction viscosity, in previous models, was given as a function of melt fraction and temperature. In contrast, we show that the viscosity variations relevant to melt focusing arise from grain-size variability and non-Newtonian creep. The asthenospheric distribution of melt fraction predicted by our models provides an improved ex- planation of the electrical resistivity structure beneath one location on the East Pacific Rise. More generally, although grain size and non-Newtonian viscosity are properties of the solid phase, we find that in the context of mid-ocean ridges, their effect on melt transport is more profound than their effect on the mantle corner-flow.Comment: 20 pages, 4 figures, 1 tabl