Unified derivation of phase-field models for alloy solidification from a grand-potential functional

In the literature, two quite different phase-field formulations for the problem of alloy solidification can be found. In the first, the material in the diffuse interfaces is assumed to be in an intermediate state between solid and liquid, with a unique local composition. In the second, the interface is seen as a mixture of two phases that each retain their macroscopic properties, and a separate concentration field for each phase is introduced. It is shown here that both types of models can be obtained by the standard variational procedure if a grand-potential functional is used as a starting point instead of a free-energy functional. The dynamical variable is then the chemical potential instead of the composition. In this framework, a complete analogy with phase-field models for the solidification of a pure substance can be established. This analogy is then exploited to formulate quantitative phase-field models for alloys with arbitrary phase diagrams. The precision of the method is illustrated by numerical simulations with varying interface thickness.Comment: 36 pages, 1 figur

Dynamics of Large-Scale Plastic Deformation and the Necking Instability in Amorphous Solids

We use the shear transformation zone (STZ) theory of dynamic plasticity to study the necking instability in a two-dimensional strip of amorphous solid. Our Eulerian description of large-scale deformation allows us to follow the instability far into the nonlinear regime. We find a strong rate dependence; the higher the applied strain rate, the further the strip extends before the onset of instability. The material hardens outside the necking region, but the description of plastic flow within the neck is distinctly different from that of conventional time-independent theories of plasticity.Comment: 4 pages, 3 figures (eps), revtex4, added references, changed and added content, resubmitted to PR

A microscopic model for solidification

We present a novel picture of a non isothermal solidification process starting from a molecular level, where the microscopic origin of the basic mechanisms and of the instabilities characterizing the approach to equilibrium is rendered more apparent than in existing approaches based on coarse grained free energy functionals \`a la Landau. The system is composed by a lattice of Potts spins, which change their state according to the stochastic dynamics proposed some time ago by Creutz. Such a method is extended to include the presence of latent heat and thermal conduction. Not only the model agrees with previous continuum treatments, but it allows to introduce in a consistent fashion the microscopic stochastic fluctuations. These play an important role in nucleating the growing solid phase in the melt. The approach is also very satisfactory from the quantitative point of view since the relevant growth regimes are fully characterized in terms of scaling exponents.Comment: 7 pages Latex +3 figures.p

Steady-state, effective-temperature dynamics in a glassy material

We present an STZ-based analysis of numerical simulations by Haxton and Liu (HL). The extensive HL data sharply test the basic assumptions of the STZ theory, especially the central role played by the effective disorder temperature as a dynamical state variable. We find that the theory survives these tests, and that the HL data provide important and interesting constraints on some of its specific ingredients. Our most surprising conclusion is that, when driven at various constant shear rates in the low-temperature glassy state, the HL system exhibits a classic glass transition, including super-Arrhenius behavior, as a function of the effective temperature.Comment: 9 pages, 6 figure

Localized induction equation and pseudospherical surfaces

We describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves, and surfaces of constant negative Gauss curvature.Comment: 21 pages, AMSTeX file. To appear in Journal of Physics A: Mathematical and Genera

Hamiltonians for curves

We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the Frenet-Serret frame of the curve. The Euler-Lagrange equations describing equilibrium are obtained; Noether's theorem is exploited to identify the constants of integration of these equations as the Casimirs of the euclidean group in three dimensions. While this system appears not to be integrable in general, it {\it is} in various limits of interest. Let the energy density be given as some function of the curvature and torsion, $f(\kappa,\tau)$. If $f$ is a linear function of either of its arguments but otherwise arbitrary, we claim that the first integral associated with rotational invariance permits the torsion $\tau$ to be expressed as the solution of an algebraic equation in terms of the bending curvature, $\kappa$. The first integral associated with translational invariance can then be cast as a quadrature for $\kappa$ or for $\tau$.Comment: 17 page

Chemical composition and origin of nebulae around Luminous Blue Variables

We use the analysis of the heavy element abundances (C, N, O, S) in circumstellar nebulae around Luminous Blue Variables to infer the evolutionary phase in which the material has been ejected. (1) We discuss the different effects that may have changed the gas composition of the nebula since it was ejected (2) We calculate the expected abundance changes at the stellar surface due to envelope convection in the red supergiant phase. If the observed LBV nebulae are ejected during the RSG phase, the abundances of the LBV nebulae require a significantly smaller amount of mass to be lost than assumed in evolutionary models. (3) We calculate the changes in the surface composition during the main sequence phase by rotation induced mixing. If the nebulae are ejected at the end of the MS-phase, the abundances in LBV nebulae are compatible with mixing times between 5 x 10^6 and 1 x 10^7 years. The existence of ON stars supports this scenario. (4) The predicted He/H ratio in the nebulae are significantly smaller than the current observed photospheric values of their central stars. Combining various arguments we show that the LBV nebulae are ejected during the blue SG phase and that the stars have not gone through a RSG phase. The chemical enhancements are due to rotation induced mixing, and the ejection is possibly triggered by near-critical rotation. During the ejection, the outflow was optically thick, which resulted in a large effective radius and a low effective temperature. This also explains the observed properties of LBV dust.Comment: 18 pages, 4 figures, to be published in The Astrophysical Journal, April 20, 200

Capillary-Wave Model for the Solidification of Dilute Binary Alloys

Starting from a phase-field description of the isothermal solidification of a dilute binary alloy, we establish a model where capillary waves of the solidification front interact with the diffusive concentration field of the solute. The model does not rely on the sharp-interface assumption, and includes non-equilibrium effects, relevant in the rapid-growth regime. In many applications it can be evaluated analytically, culminating in the appearance of an instability which, interfering with the Mullins-Sekerka instability, is similar to that, found by Cahn in grain-boundary motion.Comment: 17 pages, 12 figure

A Herschel [C II] Galactic plane survey II: CO-dark H2 in clouds

ABRIDGED: Context: HI and CO large scale surveys of the Milky Way trace the diffuse atomic clouds and the dense shielded regions of molecular hydrogen clouds. However, until recently, we have not had spectrally resolved C+ surveys to characterize the photon dominated interstellar medium, including, the H2 gas without C, the CO-dark H2, in a large sample of clouds. Aims: To use a sparse Galactic plane survey of the 1.9 THz [C II] spectral line from the Herschel Open Time Key Programme, Galactic Observations of Terahertz C+ (GOT C+), to characterize the H2 gas without CO in a statistically significant sample of clouds. Methods: We identify individual clouds in the inner Galaxy by fitting [CII] and CO isotopologue spectra along each line of sight. We combine these with HI spectra, along with excitation models and cloud models of C+, to determine the column densities and fractional mass of CO-dark H2 clouds. Results: We identify 1804 narrow velocity [CII] interstellar cloud components in different categories. About 840 are diffuse molecular clouds with no CO, 510 are transition clouds containing [CII] and 12CO, but no 13CO, and the remainder are dense molecular clouds containing 13CO emission. The CO-dark H2 clouds are concentrated between Galactic radii 3.5 to 7.5 kpc and the column density of the CO-dark H2 layer varies significantly from cloud-to-cloud with an average 9X10^(20) cm-2. These clouds contain a significant fraction of CO-dark H2 mass, varying from ~75% for diffuse molecular clouds to ~20% for dense molecular clouds. Conclusions: We find a significant fraction of the warm molecular ISM gas is invisible in HI and CO, but is detected in [CII]. The fraction of CO-dark H2 is greatest in the diffuse clouds and decreases with increasing total column density, and is lowest in the massive clouds.Comment: 21 pages, 19 figures, accepted for publication in A&A (2014

Stability of a Nonequilibrium Interface in a Driven Phase Segregating System

We investigate the dynamics of a nonequilibrium interface between coexisting phases in a system described by a Cahn-Hilliard equation with an additional driving term. By means of a matched asymptotic expansion we derive equations for the interface motion. A linear stability analysis of these equations results in a condition for the stability of a flat interface. We find that the stability properties of a flat interface depend on the structure of the driving term in the original equation.Comment: 14 pages Latex, 1 postscript-figur