Axel Voigt

A diffuse-interface method for two-phase flows with soluble surfactants

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier-Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed.

Wetting Resistance at Its Topographical Limit: The Benefit of Mushroom and Serif T Structures

Springtails (Collembola) are wingless arthropods adapted to cutaneous respiration in temporarily rain-flooded habitats. They immediately form a plastron, protecting them against suffocation upon immersion into water and even low-surface-tension liquids such as alkanes. Recent experimental studies revealed a high-pressure resistance of such plastrons against collapse. In this work, skin sections of Orthonychiurus stachianus are studied by transmission electron microscopy. The micrographs reveal cavity side-wall profiles with characteristic overhangs. These were fitted by polynomials to allow access for analytical and numerical calculations of the breakthrough pressure, that is, the barrier against plastron collapse. Furthermore, model profiles with well-defined geometries were used to set the obtained results into context and to develop a general design principle for the most robust surface structures. Our results indicate the decisive role of the sectional profile of overhanging structures to form a robust heterogeneous wetting state for low-surface-tension liquids that enables the omniphobicity. Furthermore, the design principles of mushroom and serif T structures pave the way for omniphobic surfaces with a high-pressure resistance irrespective of solid surface chemistry.

A new phase-field model for strongly anisotropic systems

We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn–Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn–Hilliard equations become ill-posed. Our models contain a high-order Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularized equations are sixth order in space. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergence of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-field model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytical sharp-interface theory that describes the rounding of the sharp corners by the Willmore regularization.

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.

Surface evolution of elastically stressed films under deposition by a diffuse interface model

We consider the heteroepitaxial growth of thin films by numerical simulations within a diffuse interface model. The model is applicable to describe the self-organization of nanostructures. The influence of strain, surface energies and kinetics on the surface evolution is considered. A matched asymptotic analysis shows the formal convergence of an anisotropic viscous Cahn-Hilliard model to a general surface evolution equation. The system is solved by adaptive finite elements in three dimensions and in special cases compared with sharp interface models.

Dynamics of multicomponent vesicles in a viscous fluid

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibil-ity of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.

Nucleation and growth by a phase field crystal (PFC) model

We review the derivation of a phase field crystal (PFC) model from classical density functional theory (DFT). Through a gradient flow of the Helmholtz free energy functional and appropriate approximations of the correlation functions, higher order nonlinear equations are derived for the evolution of a time averaged density. The equation is solved by finite elements using a semi-implicit time discretization.

A continuum model of colloid-stabilized interfaces

Colloids that are partially wetted by two immiscible fluids can become confined to fluid-fluidinterfaces. At sufficiently high volume fractions, the colloids may jam and the interface may crystallize. Examples include bicontinuous interfacially jammed emulsion gels (bijels), which were proposed in this study by Stratford et al. [Science 309, 2198 (2005)] as a hypothetical new class of soft materials in which interpenetrating, continuous domains of two immiscible viscous fluids are maintained in a rigid state by a jammed layer of colloidal particles at their interface. We develop a continuum model for such a system that is capable of simulating the long-time evolution. A Navier-Stokes-Cahn-Hilliard model for the macroscopic two-phase flow system is combined with a surface phase-field-crystal model for the microscopic colloidal system along the interface. The presence of colloids introduces elastic forces at the interface between the two immiscible fluid phases. An adaptive finite element method is used to solve the model numerically. Using a variety of flow configurations in two dimensions, we demonstrate that as colloids jam on the interface and the interfacecrystallizes, the elastic force may be strong enough to make the interface sufficiently rigid to resist external forces, such as an applied shear flow, as well as surface tension induced coarsening in bicontinuous structures.

Software concepts and numerical algorithms for a scalable adaptive parallel finite element method

An efficient implementation of an adaptive finite element method on distributed memory systems requires an efficient linear solver. Most solver methods, which show scalability to a large number of processors make use of some geometric information of the mesh. This information has to be provided to the solver in an efficient and solver specific way. We introduce data structures and numerical algorithms which fulfill this task and allow in addition for an user-friendly implementation of a large class of linear solvers. The concepts and algorithms are demonstrated for global matrix solvers and domain decomposition methods for various problems in fluid dynamics, continuum mechanics and materials science. Weak and strong scaling is shown for up to 16.384 processors.

A diffuse-interface approximation for step flow in epitaxial growth

We consider a step-flow model for epitaxial growth, as proposed by Burton et al. This type of model is discrete in the growth direction but continuous in the lateral directions. The effect of the Ehrlich–Schwoebel barrier, which limits the attachment rate of adatoms to a step from an upper terrace, is included. Mathematically, this model is a 2+1-dimensional dynamic free boundary problem for the steps.In this paper, we propose a diffuse-interface approximation which reproduces an arbitrary Ehrlich–Schwoebel barrier. This is achieved by introducing a degenerate mobility into the so-called viscous Cahn–Hilliard equation. We relate this modified Cahn–Hilliard equation to the sharp interface model via formal matched asymptotic expansion.