arXiv subjects

Ingo Nitschke

Publications and source records attributed to Ingo Nitschke.

At least 19 recordsLinked to original sources

arXiv2026

Surface Phase-Field-Crystal-Helfrich model for out-of-plane deformations in thin crystalline sheets with lattice mismatch

Thin, flexible crystalline sheets exhibit unique elastic properties due to their ability to undergo out-of-plane deformations. Understanding this behavior requires a description that couples in-plane elasticity, out-of-plane deformation, and their coupling, taking the crystalline structure and its defects into account. We develop a multiscale description for these systems by extending the surface Phase-Field-Crystal-Helfrich model. The extension permits a spatially varying equilibrium lattice spacing, enabling the representation of localized lattice eigenstrain to mimic lattice mismatch in heterostructures. We validate the extended model against analytical predictions from classical F\"oppl-von K\'arm\'an equations for uniaxial compression and from Eshelby's inclusion problem. Using this validated framework, we then show how locally induced compressive stresses drive out-of-plane deformation in the sheets.

cond-mat.mtrl-sci↗

arXiv2026

Scalar Truesdell Time Derivative and $(L^{2},H^{-1})$ -- Surface Gradient Flows

We address surface gradient flows which allow for energy dissipation by evolving the surface and a scalar quantity on it, simultaneously. A proper choice of the time derivative and the gauge of surface independence guarantees energy dissipation and ensures conservation of the scalar quantity. The resulting system of partial differential equations couples geometric evolution equations for the evolution of the surface in normal directions, equations for tangential movement and scalar-valued equations on the evolving surface. We discuss the general setting and the special case of surface tension flows and numerically demonstrate the importance of tangential movement on the evolution.

math-ph↗

arXiv2025

Hydrodynamic liquid crystal models for lipid bilayers

Coarse-grained continuous descriptions for lipid bilayers are typically based on minimizing the Helfrich energy. Such models consider the fluid properties of these structures only implicitly and have been shown to nicely reproduce equilibrium properties. Model extensions that also address the dynamics of these structures are surface (Navier--)Stokes--Helfrich models. They explicitly account for membrane viscosity. However, these models also usually treat the lipid bilayer as a homogeneous continuum, neglecting the molecular degrees of freedom of the lipids. Here, we derive refined models which consider in addition a scalar order parameter representing the molecular alignment of the lipids along the surface normal. Starting from hydrodynamic surface liquid crystal models, we obtain a hydrodynamic surface Landau--Helfrich model for asymmetric lipid bilayers and a surface Beris--Edwards model for symmetric lipid bilayers. The fully ordered case for both models leads to the known surface (Navier--)Stokes--Helfrich models. Besides more detailed continuous models for lipid bilayers, we therefore also provide an alternative derivation of surface (Navier--)Stokes--Helfrich models.

cond-mat.soft↗

arXiv2024

The influence of higher order geometric terms on the asymmetry and dynamics of membranes

We consider membranes as fluid deformable surface and allow for higher order geometric terms in the bending energy. The evolution equations are derived and numerically solved using surface finite elements. The higher order geometric terms related to the Gaussian curvature squared have a tendency to stabilize tubes and enhance the evolution towards equilibrium shapes, thereby facilitating rapid shape changes. This is demonstrated in axisymmetric settings and fully three-dimensional simulations.

cond-mat.soft↗

arXiv2024

Active nematodynamics on deformable surfaces

We consider active nematodynamics on deformable surfaces. Based on a thermodynamically consistent surface Beris-Edwards model we add nematic activity and focus on the emerging additional coupling mechanism between the nematic field, the flow field and the curved surface. We analyse the impact of the active nematic force at topological defects. Under the presence of curvature all defects become active and contribute not only tangential forces but also normal forces. This confirms the proposed role of topological defects in surface evolution and provides the basis for a dynamic description of morphogenetic processes.

cond-mat.soft↗

arXiv2023

Beris-Edwards Models on Evolving Surfaces: A Lagrange-D'Alembert Approach

Using the Lagrange-D'Alembert principle we develop thermodynamically consistent surface Beris-Edwards models. These models couple viscous inextensible surface flow with a Landau-de Gennes-Helfrich energy and consider the simultaneous relaxation of the surface Q-tensor field and the surface, by taking hydrodynamics of the surface into account. We consider different formulations, a general model with three-dimensional surface Q-tensor dynamics and possible constraints incorporated by Lagrange multipliers and a surface conforming model with tangential anchoring of the surface Q-tensor field and possible additional constraints. In addition to different treatments of the surface Q-tensor, which introduces different coupling mechanisms with the geometric properties of the surface, we also consider different time derivatives to account for different physical interpretations of surface nematics. We relate the derived models to established models in simplified situations, compare the different formulations with respect to numerical realizations and mention potential applications in biology.

math-ph↗

arXiv2023

Mesoscale modeling of deformations and defects in thin crystalline sheets

We present a mesoscale description of deformations and defects in thin, flexible sheets with crystalline order, tackling the interplay between in-plane elasticity, out-of-plane deformation, as well as dislocation nucleation and motion. Our approach is based on the Phase-Field Crystal (PFC) model, which describes the microscopic atomic density in crystals at diffusive timescales, naturally encoding elasticity and plasticity effects. In its amplitude expansion (APFC), a coarse-grained description of the mechanical properties of crystals is achieved. We introduce surface PFC and surface APFC models in a convenient height-function formulation encoding deformation in the normal direction. This framework is proven consistent with classical aspects of strain-induced buckling, defect nucleation on deformed surfaces, and out-of-plane relaxation near dislocations. In particular, we benchmark and discuss the results of numerical simulations by looking at the continuum limit for buckling under uniaxial compression and at evidence from microscopic models for deformation at defects and defect arrangements, demonstrating the scale-bridging capabilities of the proposed framework. Results concerning the interplay between lattice distortion at dislocations and out-of-plane deformation are also illustrated by looking at the annihilation of dislocation dipoles and systems hosting many dislocations. With the novel formulation proposed here, and its assessment with established approaches, we envision applications to multiscale investigations of crystalline order on deformable surfaces.

cond-mat.mtrl-sci↗

arXiv2023

Derivation and simulation of a two-phase fluid deformable surface model

We consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modeled by incompressible surface Navier-Stokes-Cahn-Hilliard-like equations with bending forces. We derive this model using the Lagrange-D'Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor-Hood element for the surface Navier-Stokes part, a semi-implicit approach for the Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified sub-problems.

math.NA↗

arXiv2023

Tensorial time derivatives on moving surfaces: General concepts and a specific application for surface Landau-de Gennes models

Observer-invariance is regarded as a minimum requirement for an appropriate definition of time derivatives. We systematically discuss such time derivatives for surface tensor field and provide explicit formulations for material, upper-convected, lower-convected and Jaumann/corotational time derivatives which all lead to different physical implications. We compare these results with the corresponding time derivatives for tangential tensor fields. As specific surface 2-tensor fields we consider surface Q-tensor fields and conforming surface Q-tensor fields and apply the results in surface Landau-de Gennes models for surface liquid crystals.

math-ph↗

arXiv2022

Tangential Tensor Fields on Deformable Surfaces -- How to Derive Consistent $L^2$-Gradient Flows

We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.

math-ph↗

arXiv2020

Observer-invariant time derivatives on moving surfaces

Observer-invariance is regarded as a minimum requirement for an appropriate definition and derived systematically from a spacetime setting, where observer-invariance is a special case of a covariance principle and covered by Ricci-calculus. The analysis is considered for tangential n-tensor fields on moving surfaces and provides formulations which are applicable for computations. For various special cases, e.g., vector fields (n = 1) and symmetric and trace-less tensor fields (n = 2) we compare material and convected derivatives and demonstrate the different underlying physics.

math-ph↗

arXiv2020

A numerical approach for fluid deformable surfaces

Fluid deformable surfaces show a solid-fluid duality which establishes a tight interplay between tangential flow and surface deformation. We derive the governing equations as a thin film limit and provide a general numerical approach for their solution. The simulation results demonstrate the rich dynamics resulting from this interplay, where in the presence of curvature any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flow. However, they also show that the only possible stable stationary state in the considered setting is a sphere with zero velocity.

physics.comp-ph↗

arXiv2019

Properties of surface Landau-de Gennes Q-tensor models

Uniaxial nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface offer a non trivial interplay between the geometry and the topology of the surface and the orientational degree of freedom. We consider a general thin film limit of a Landau-de Gennes Q-tensor model which retains the characteristics of the 3D model. From this, previously proposed surface models follow as special cases. We compare fundamental properties, such as alignment of the orientational degrees of freedom with principle curvature lines, order parameter symmetry and phase transition type for these models, and suggest experiments to identify proper model assumptions.

cond-mat.soft↗

arXiv2019

Liquid Crystals on Deformable Surfaces

Liquid crystals with molecules constrained to the tangent bundle of a curved surface show interesting phenomena resulting from the tight coupling of the elastic and bulk free energies of the liquid crystal with geometric properties of the surface. We derive thermodynamically consistent Frank-Oseen-Helfrich and Landau-de Gennes-Helfrich models which consider the simultaneous relaxation of the director/Q-tensor fields and the surface. The resulting systems of vector- or tensor-valued surface partial differential equation and geometric evolution laws are numerically solved to tackle the rich dynamics of these systems and to compute the resulting equilibrium shapes. The results strongly depend on the intrinsic and extrinsic curvature contributions and can lead to unexpected asymmetric shapes.

cond-mat.soft↗

arXiv2019

Comment on Geometrical Control of Active Turbulence in Curved Topographies

In the recent letter Pearce et. al. Phys. Rev. Lett. 122, 168002 (2019) the authors investigate the turbulent dynamics of a two-dimensional active nematic liquid crystal which is constrained to the surface of a torus. The underlying model combines an incompressible surface Navier-Stokes equation with friction and active forcing with a surface Landau-de Gennes model for nematic liquid crystals. To numerically solve the surface Navier-Stokes equation a vorticity-stream function approach is considered. This approach is not appropriate on surfaces that are not simply connected - such as the considered torus - due to non-trivial harmonic vector fields. We will explain the underlying situation and provide details and examples to rebut the argumentation in Pearce et. al. Phys. Rev. Lett. 122, 168002 (2019).

cond-mat.soft↗

arXiv2019

Hydrodynamic interactions in polar liquid crystals on evolving surfaces

We consider the derivation and numerical solution of the flow of passive and active polar liquid crystals, whose molecular orientation is subjected to a tangential anchoring on an evolving curved surface. The underlying passive model is a simplified surface Ericksen-Leslie model, which is derived as a thin-film limit of the corresponding three-dimensional equations with appropriate boundary conditions. A finite element discretization is considered and the effect of hydrodynamics on the interplay of topology, geometric properties and defect dynamics is studied for this model on various stationary and evolving surfaces. Additionally, we consider an active model. We propose a surface formulation for an active polar viscous gel and exemplarily demonstrate the effect of the underlying curvature on the location of topological defects on a torus.

physics.comp-ph↗

arXiv2018

A finite element approach for vector- and tensor-valued surface PDEs

We derive a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on general manifolds. This allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surface PDEs. We consider piecewise linear Lagrange surface finite elements on triangulated surfaces and validate the approach by a vector- and a tensor-valued surface Helmholtz problem on an ellipsoid. We experimentally show optimal (linear) order of convergence for these problems. The full functionality is demonstrated by solving a surface Landau-de Gennes problem on the Stanford bunny. All tools required to apply this approach to other vector- and tensor-valued surface PDEs are provided.

math.NA↗

arXiv2017

Nematic liquid crystals on curved surfaces - a thin film limit

We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.

cond-mat.soft↗