Elimination of MHD current sheets by modifications to the plasma wall in a fixed boundary model.

Models of magnetohydrodynamic (MHD) equilibia that for computational convenience assume the existence of a system of nested magnetic flux surfaces tend to exhibit singular current sheets. These sheets are located on resonant flux surfaces that are associated with rational values of the rotational transform. We study the possibility of eliminating these singularities by suitable modifications of the plasma boundary, which we prescribe in a fixed boundary setting. We find that relatively straightforward iterative procedures can be used to eliminate weak current sheets that are generated at resonant flux surfaces by the nonlinear interactions of resonating wall harmonics. These types of procedures may prove useful in the design of fusion devices with configurations that enjoy improved stability and transport properties.

Effect of vacancy creation and annihilation on grain boundary motion.

Interaction of vacancies with grain boundaries (GBs) is involved in many processes occurring in materials, including radiation damage healing, diffusional creep, and solid-state sintering. We analyze a model describing a set of processes occurring at a GB in the presence of a non-equilibrium, non-homogeneous vacancy concentration. Such processes include vacancy diffusion toward, away from, and across the GB, vacancy generation and absorption at the GB, and GB migration. Numerical calculations within this model reveal that the coupling among the different processes gives rise to interesting phenomena, such as vacancy-driven GB motion and accelerated vacancy generation/absorption due to GB motion. The key combinations of the model parameters that control the kinetic regimes of the vacancy-GB interactions are identified via a linear stability analysis. Possible applications and extensions of the model are discussed.

Model reduction of rigid-body molecular dynamics via generalized multipole potentials.

Motivated by the challenges of uncertainty quantification for coarse-grained (CG) molecular dynamics, we investigate the role of perturbation theory in model reduction of classical systems. In particular, we consider the task of coarse-graining rigid bodies in the context of generalized multipole potentials that have controllable levels of accuracy relative to their atomistic counterparts. We show how the multipole framework yields a hierarchy of models that systematically connects a CG "point molecule" approximation to the exact dynamics. We use these results to understand when and how the CG models fail to describe atomistic dynamics at the trajectory level and develop asymptotic error estimates for approximate molecular potential energies. Implications for other model-reduction strategies are also discussed. Key findings of this work are that (i) omitting rotational energy introduces significant error when coarse-graining and (ii) attention to symmetry can improve accuracy of "point-molecule" approximations. Analytical derivations and numerical results support these conclusions. Relevance to nonrigid bodies is also discussed.

Numerical Analysis and Simulation for a Generalized Planar Ginzburg-Landau Equation in a Circular Geometry.

In this paper, a numerical scheme for a generalized planar Ginzburg-Landau energy in a circular geometry is studied. A spectral-Galerkin method is utilized, and a stability analysis and an error estimate for the scheme are presented. It is shown that the scheme is unconditionally stable. We present numerical simulation results that have been obtained by using the scheme with various sets of boundary data, including those the form u(θ) = exp(idθ), where the integer d denotes the topological degree of the solution. These numerical results are in good agreement with the experimental and analytical results. Results include the computation of bifurcations from pure bend or splay patterns to spiral patterns for d = 1, and computations of metastable or unstable higher-energy solutions as well as the lowest energy ground state solutions for values of d ranging from two to five.

On the Stability of Rotating Drops.

We consider the equilibrium and stability of rotating axisymmetric fluid drops by appealing to a variational principle that characterizes the equilibria as stationary states of a functional containing surface energy and rotational energy contributions, augmented by a volume constraint. The linear stability of a drop is determined by solving the eigenvalue problem associated with the second variation of the energy functional. We compute equilibria corresponding to both oblate and prolate shapes, as well as toroidal shapes, and track their evolution with rotation rate. The stability results are obtained for two cases: (i) a prescribed rotational rate of the system ("driven drops"), or (ii) a prescribed angular momentum ("isolated drops"). For families of axisymmetric drops instabilities may occur for either axisymmetric or non-axisymmetric perturbations; the latter correspond to bifurcation points where non-axisymmetric shapes are possible. We employ an angle-arc length formulation of the problem which allows the computation of equilibrium shapes that are not single-valued in spherical coordinates. We are able to illustrate the transition from spheroidal drops with a strong indentation on the rotation axis to toroidal drops that do not extend to the rotation axis. Toroidal drops with a large aspect ratio (major radius to minor radius) are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. Prolate spheroidal shapes occur if a drop of lower density rotates within a denser medium; these drops appear to be linearly stable. This work is motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013].

Equilibrium and stability of axisymmetric drops on a conical substrate under gravity.

Motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013], we study a related problem of the determination of the equilibrium and stability of axisymmetric drops on a conical substrate in the presence of gravity. A variational principle is used to characterize equilibrium shapes that minimize surface energy and gravitational potential energy subject to a volume constraint, and the resulting Euler equation is solved numerically using an angle/arclength formulation. The resulting equilibria satisfy a Laplace-Young boundary condition that specifies the contact angle at the three-phase trijunction. The vertical position of the equilibrium drops on the cone is found to vary significantly with the dimensionless Bond number that represents the ratio of gravitational and capillary forces; a global force balance is used to examine the conditions that affect the drop positions. In particular, depending on the contact angle and the cone half-angle, we find that the vertical position of the drop can either increase ("the drop climbs the cone") or decrease due to a nominal increase in the gravitational force. Most of the equilibria correspond to upward-facing cones, and are analogous to sessile drops resting on a planar surface; however we also find equilibria that correspond to downward facing cones, that are instead analogous to pendant drops suspended vertically from a planar surface. The linear stability of the drops is determined by solving the eigenvalue problem associated with the second variation of the energy functional. The drops with positive Bond number are generally found to be unstable to non-axisymmetric perturbations that promote a tilting of the drop. Additional points of marginal stability are found that correspond to limit points of the axisymmetric base state. Drops that are far from the tip are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. We have also found a range of completely stable solutions that correspond to small contact angles and cone half-angles.

Instability in pipe flow.

The long-puzzling, unphysical result that linear stability analyses lead to no transition in pipe flow, even at infinite Reynolds number, is ascribed to the use of stick boundary conditions, because they ignore the amplitude variations associated with the roughness of the wall. Once that length scale is introduced (here, crudely, through a corrugated pipe), linear stability analyses lead to stable vortex formation at low Reynolds number above a finite amplitude of the corrugation and unsteady flow at a higher Reynolds number, where indications are that the vortex dislodges. Remarkably, extrapolation to infinite Reynolds number of both of these transitions leads to a finite and nearly identical value of the amplitude, implying that below this amplitude, the vortex cannot form because the wall is too smooth and, hence, stick boundary results prevail.

Convective Instabilities in Two Liquid Layers.

We perform linear stability calculations for horizontal fluid bilayers, taking into account both buoyancy effects and thermocapillary effects in the presence of a vertical temperature gradient. To help understand the mechanisms driving the instability, we have performed both long-wavelength and short-wavelength analyses. The mechanism for the large wavelength instability is complicated, and the detailed form of the expansion is found to depend on the Crispation and Bond numbers. The system also allows a conventional Rayleigh-Taylor instability if heavier fluid overlies lighter fluid, and the long-wavelength analysis describes this case as well. In addition to the asymptotic analyses for large and small wavelengths, we have performed numerical calculations using materials parameters for a benzene-water system.

Phase field modeling of electrochemistry. I. Equilibrium.

A diffuse interface (phase field) model for an electrochemical system is developed. We describe the minimal set of components needed to model an electrochemical interface and present a variational derivation of the governing equations. With a simple set of assumptions: mass and volume constraints, Poisson's equation, ideal solution thermodynamics in the bulk, and a simple description of the competing energies in the interface, the model captures the charge separation associated with the equilibrium double layer at the electrochemical interface. The decay of the electrostatic potential in the electrolyte agrees with the classical Gouy-Chapman and Debye-Hückel theories. We calculate the surface free energy, surface charge, and differential capacitance as functions of potential and find qualitative agreement between the model and existing theories and experiments. In particular, the differential capacitance curves exhibit complex shapes with multiple extrema, as exhibited in many electrochemical systems.

Phase field modeling of electrochemistry. II. Kinetics.

The kinetic behavior of a phase field model of electrochemistry is explored for advancing (electrodeposition) and receding (electrodissolution) conditions in one dimension. We previously described the equilibrium behavior of this model [J. E. Guyer, W. J. Boettinger, J. A. Warren, and G. B. McFadden, Phys. Rev. E 69, 021603 (2004)]. We examine the relationship between the parameters of the phase field method and the more typical parameters of electrochemistry. We demonstrate ohmic conduction in the electrode and ionic conduction in the electrolyte. We find that, despite making simple, linear dynamic postulates, we obtain the nonlinear relationship between current and overpotential predicted by the classical "Butler-Volmer" equation and observed in electrochemical experiments. The charge distribution in the interfacial double layer changes with the passage of current and, at sufficiently high currents, we find that the diffusion limited deposition of a more noble cation leads to alloy deposition with less noble species.

Steps, kinetic anisotropy, and long-wavelength instabilities in directional solidification.

We consider the effect of anisotropic interface kinetics on long-wavelength instabilities during the directional solidification of a binary alloy having a vicinal interface. Linear theory predicts that a planar solidification front is stabilized under the effect of anisotropy as long as the segregation coefficient is small enough, whereas a novel instability appears at high rates of solidification. Furthermore, the neutral stability curve, indicating the values of the principal control parameter (here the morphological number) for which the growth rate of a sinusoidal perturbation of a given wavelength changes its sign, is shown to have up to three branches, two of them combining to form an isola for certain values of the control parameters. We identify conditions for which linear stability theory predicts the instability of the planar interface to long-wavelength traveling waves. A number of distinguished limits provide evolution equations that describe the resulting dynamical behavior of the crystal-melt interface and generalize previous work by Sivashinsky, Brattkus, and Davis and Riley and Davis. Bifurcation analysis and numerical computations for the derived evolution equations show that the anisotropy is able to promote the tendency to supercritical bifurcation, and also leads to the development of strongly preferred interface orientations for finite-amplitude deformations.

Some perspectives on the interpretation of proton NMR spin diffusion data in terms of polymer morphologies.

Proton spin diffusion data yield morphological information over dimensions covering approximately the 2-50 nm range. In this article, the interpretation of such data for polymers is emphasized, recognizing that the mathematical framework for much of this interpretation already exists in the literature. Practical issues are considered, for example, a useful scaling of plotted data is suggested, key attributes of the data are identified and ambiguities in the mapping of data into morphological models are spelled out. Discussion is limited to two-phase systems, where it is assumed that, by employing multiple-pulse methods polarization gradients can be generated, whose spunal sharpness is limited solety by the morphological definition of the interfaces. Interpretation of data in terms of morphology and stoichiometry is emphasized, where stoichiometric issues pertain only to chemically heterogeneous systems. Extraction of stoichiometric information from spin diffusion data is not commonly attempted; the discussion included herein allows for the possibility that the composition of phases may be chemically mixed. Methods for generating gradients are discussed only briefly. A standardized spin diffusion plot is proposed and the initial slope of this plot is tocussed on for providing information about morphology and stoichiometry. Ambiguities of interpretation considered include the dimensionality of the deduced morphology and, for systems with chemical heterogeneity the uniqueness of the compositional characterization of each phase. In addition, funite difference methods are used to simulate entire spin diffusion curves for idealized lamellar and hexagonal rod/matrix morphologies. Comparisons of these curves show that distinguishing 1-D and 2-D morphologies on the basis of experimental data is unlikely to be successful over the range of stoichiometrics where such morphologies are expected. Several examples of spin diffusion data are presented. Brief treatments of the following topics are included: finite interface width, estimation of spin diffusion constants, and incorporation of longitudinal relaxation effects. Finally, a short experimental discussion on the preparation of polarization gradients is given including those preparations which make use of differences in the multiple-pulse relaxation time, T1xz. It is noted that T1xz decays may be strongly perturbed in the presence of magic angle spinning, therefore, strategies are also outlined for minimizing these effects.