arXiv subjects

Axel Voigt

Publications and source records attributed to Axel Voigt.

102 records · Page 6Linked to original sources

arXiv2015

Liquid crystalline growth within a phase-field crystal model

By using a phase-field crystal (PFC) model, the liquid-crystal growth of the plastic triangular phase is simulated with emphasis on crystal shape and topological defect formation. The equilibrium shape of a plastic triangular crystal (PTC) grown from a isotropic phase is compared with that grown from a columnar/smectic A (CSA) phase. While the shape of a PTC nucleus in the isotropic phase is almost identical to that of a classical PFC model, the shape of a PTC nucleus in CSA is affected by the orientation of stripes in the CSA phase, and irregular hexagonal, elliptical, octagonal, and rectangular shapes are obtained. Concerning the dynamics of the growth process we analyse the topological structure of the nematic-order, which starts from nucleation of $+\frac{1}{2}$ and $-\frac{1}{2}$ disclination pairs at the PTC growth front and evolves into hexagonal cells consisting of $+1$ vortices surrounded by six satellite $-\frac{1}{2}$ disclinations. It is found that the orientational and the positional order do not evolve simultaneously, the orientational order evolves behind the positional order, leading to a large transition zone, which can span over several lattice spacings.

cond-mat.soft↗

arXiv2014

The interplay of curvature and vortices in flow on curved surfaces

Incompressible fluids on curved surfaces are considered with respect to the interplay between topology, geometry and fluid properties using a surface vorticity-stream function formulation, which is solved using parametric finite elements. Motivated by designed examples for superfluids, we consider the influence of a geometric potential on vortices for fluids with finite viscosity and show numerical examples in which a change in the geometry is used to manipulate the flow field.

physics.flu-dyn↗

arXiv2014

Detached topological charge on capillary bridges

We numerically investigate crystalline order on negative Gaussian curvature capillary bridges. In agreement with the experimental results in [W. Irvine et al., Nature, "Pleats in crystals on curved surfaces", 2010, (468), 947]} we observe for decreasing integrated Gaussian curvature a sequence of transitions, from no defects to isolated dislocations, pleats, scars and isolated sevenfold disclinations. We especially focus on the dependency of the detached topological charge on the integrated Gaussian curvature, for which we observe, again in agreement with the experimental results, no net disclination for an integrated curvature down to -10, and a linear behaviour from there on until the disclinations match the integrated curvature of -12. The results are obtained using a phase field crystal approach on catenoid-like surfaces and are highly sensitive to the initialization.

cond-mat.soft↗

arXiv2013

Diffuse interface models of locally inextensible vesicles in a viscous fluid

We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. To solve for the local Lagrange multiplier, we introduce a new equation whose solution essentially provides a harmonic extension of the local Lagrange multiplier off the interface while maintaining the local inextensibility constraint near the interface. To make the method more robust, we develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. Asymptotic analysis is presented that shows that our new system converges to a relaxed version of the inextensible sharp interface model. This is also verified numerically. Although the model does not depend on dimension, we present numerical simulations only in 2D. To solve the 2D equations numerically, we develop an efficient algorithm combining an operator splitting approach with adaptive finite elements where the Navier-Stokes equations are implicitly coupled to the diffuse interface inextensibility equation. Numerical simulations of a single vesicle in a shear flow at different Reynolds numbers demonstrate that errors in enforcing local inextensibility may accumulate and lead to large differences in the dynamics in the tumbling regime and differences in the inclination angle of vesicles in the tank-treading regime. The local relaxation algorithm is shown to effectively prevent this accumulation by driving the system back to its equilibrium state when errors in local inextensibility arise.

math-ph↗

arXiv2013

Grain growth beyond the Mullins model, capturing the complex physics behind universal grain size distributions

Grain growth experiments on thin metallic films have shown the geometric and topological characteristics of the grain structure to be universal and independent of many experimental conditions. The universal size distribution, however, is found to differ both qualitatively and quantitatively from the standard Mullins curvature driven model of grain growth; with the experiments exhibiting an excess of small grains (termed an "ear") and an excess of very large grains (termed a "tail") compared with the model. While a plethora of extensions of the Mullins model have been proposed to explain these characteristics, none have been successful. In this work, large scale simulations of a model that resolves the atomic scale on diffusive time scales, the phase field crystal model, is used to examine the complex phenomena of grain growth. The results are in remarkable agreement with the experimental results, recovering the characteristic "ear" and "tail" features of the experimental grain size distribution. The simulations also indicate that while the geometric and topological characteristics are universal, the dynamic growth exponent is not.

cond-mat.mtrl-sci↗

arXiv2013

Structure and dynamics of interfaces between two coexisting liquid crystalline phases

The phase-field-crystal model is used to access the structure and thermodynamics of interfaces between two coexisting liquid crystalline phases in two spatial dimensions. Depending on the model parameters there is a variety of possible coexistences between two liquid crystalline phases including a plastic triangular crystal (PTC). Here, we calculate numerically the profiles for the mean density and the nematic order tensor across the interface for isotropic-PTC and columnar-PTC respectively smectic A-PTC coexistence. As a general finding, the width of the interface with respect to the nematic order parameter characterizing the orientational order is larger than the width of the mean density interface. In approaching the interface from the PTC side, at first the mean density goes down and then the nematic order parameter follows. The relative shift of the two profiles can be larger than a full lattice constant of the plastic crystal. Finally, we also present numerical results for the dynamic relaxation of an initial order-parameter profile towards its equilibrium interfacial profile. Our predictions for the interfacial profiles can in principle be verified in real-space experiments of colloidal dispersions.

cond-mat.soft↗

arXiv2010

A multi-mesh finite element method for Lagrange elements of arbitrary degree

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid-solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier-Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.

math.NA↗

arXiv2010

Particles on curved surfaces - a dynamic approach by a phase field crystal model

We present a dynamic model to study ordering of particles on arbitrary curved surfaces. Thereby the particles are represented as maxima in a density field and a surface partial differential equation for the density field is solved to the minimal energy configuration. We study annihilation of dislocations within the ordered sytem and premelting along grain boundary scars. The obtained minimal energy configurations on a sphere are compared with existing results and scaling laws are computed for the number of excess dislocations as a function of system size.

cond-mat.mtrl-sci↗

arXiv2010

A phase-field-crystal approach to critical nuclei

We investigate a phase-field-crystal model for homogeneous nucleation. Instead of solving the time evolution of a density field towards equilibrium we use a String Method to identify saddle points in phase space. The saddle points allow to obtain the nucleation barrier and the critical nucleus. The advantage of using the phase-field-crystal model for this task is its ability to resolve atomistic effects. The obtained results indicate different properties of the critical nucleus compared with bulk crystals and show a detailed description of the nucleation process.

cond-mat.stat-mech↗

arXiv2009

Derivation of the phase field crystal model for colloidal solidification

The phase-field crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density functional theory. The different underlying approximations are discussed. In particular, a variant of the phase-field crystal model is proposed which involves less approximations than the standard phase-field crystal model. We finally test the validity of these phase-field crystal models against dynamical density functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field crystal model in order to match the bulk freezing transition point.

cond-mat.soft↗

arXiv2007

Morphological stability of electromigration-driven vacancy islands

The electromigration-induced shape evolution of two-dimensional vacancy islands on a crystal surface is studied using a continuum approach. We consider the regime where mass transport is restricted to terrace diffusion in the interior of the island. In the limit of fast attachment/detachment kinetics a circle translating at constant velocity is a stationary solution of the problem. In contrast to earlier work [O. Pierre-Louis and T.L. Einstein, Phys. Rev. B 62, 13697 (2000)] we show that the circular solution remains linearly stable for arbitrarily large driving forces. The numerical solution of the full nonlinear problem nevertheless reveals a fingering instability at the trailing end of the island, which develops from finite amplitude perturbations and eventually leads to pinch-off. Relaxing the condition of instantaneous attachment/detachment kinetics, we obtain non-circular elongated stationary shapes in an analytic approximation which compares favorably to the full numerical solution.

cond-mat.mtrl-sci↗

arXiv2005

Complex shape evolution of electromigration-driven single-layer islands

The shape evolution of two-dimensional islands through periphery diffusion biased by an electromigration force is studied numerically using a continuum approach. We show that the introduction of crystal anisotropy in the mobility of edge atoms induces a rich variety of migration modes, which include oscillatory and irregular behavior. A phase diagram in the plane of anisotropy strength and island size is constructed. The oscillatory motion can be understood in terms of stable facets which develop on one side of the island and which the island then slides past. The facet orientations are determined analytically.

cond-mat.mtrl-sci↗