arXiv subjects

Martin Michael Mueller

Publications and source records attributed to Martin Michael Mueller.

9 recordsLinked to original sources

arXiv2013

Dipoles in thin sheets

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.

cond-mat.soft↗

arXiv2012

Morphogenesis of membrane invaginations in spherical confinement

We study the morphology of a fluid membrane in spherical confinement. When the area of the membrane is slightly larger than the area of the outer container, a single axisymmetric invagination is observed. For higher area, self-contact occurs: the invagination breaks symmetry and deforms into an ellipsoid-like shape connected to its outer part via a small slit. For even higher areas, a second invagination forms inside the original invagination. The folding patterns observed could constitute basic building blocks in the morphogenesis of biological tissues and organelles.

physics.bio-ph↗

arXiv2011

Conical instabilities on paper

The stability of the fundamental defects of an unstretchable flat sheet is examined. This involves expanding the bending energy to second order in deformations about the defect. The modes of deformation occur as eigenstates of a fourth-order linear differential operator. Unstretchability places a global linear constraint on these modes. Conical defects with a surplus angle exhibit an infinite number of states. If this angle is below a critical value, these states possess an n-fold symmetry labeled by an integer, n \geq 2. A nonlinear stability analysis shows that the 2-fold ground state is stable, whereas excited states possess 2(n - 2) unstable modes which come in even and odd pairs.

cond-mat.soft↗

arXiv2008

Conical defects in growing sheets

A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle $se$ at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if $se <= 0$, the disc can fold into one of a discrete infinite number of states if $se$ is positive. We construct these states in the regime where bending dominates, determine their energies and how stress is distributed in them. For each state a critical value of $se$ is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has two-fold symmetry.

cond-mat.other↗

arXiv2007

How paper folds: bending with local constraints

A variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry. This framework is applied to describe the patterns of a folded sheet of paper. The unstretchability of paper implies a constraint on the surface metric; bending is penalized by an energy quadratic in mean curvature. The local Lagrange multipliers enforcing the constraint are identified with a conserved tangential stress that couples to the extrinsic curvature of the sheet. The framework is illustrated by examining the deformation of a flat sheet into a generalized cone.

cond-mat.other↗

arXiv2007

Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations

Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along any contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.

cond-mat.soft↗

arXiv2006

How to determine local elastic properties of lipid bilayer membranes from atomic-force-microscope measurements: A theoretical analysis

Measurements with an atomic force microscope (AFM) offer a direct way to probe elastic properties of lipid bilayer membranes locally: provided the underlying stress-strain relation is known, material parameters such as surface tension or bending rigidity may be deduced. In a recent experiment a pore-spanning membrane was poked with an AFM tip, yielding a linear behavior of the force-indentation curves. A theoretical model for this case is presented here which describes these curves in the framework of Helfrich theory. The linear behavior of the measurements is reproduced if one neglects the influence of adhesion between tip and membrane. Including it via an adhesion balance changes the situation significantly: force-distance curves cease to be linear, hysteresis and nonzero detachment forces can show up. The characteristics of this rich scenario are discussed in detail in this article.

cond-mat.soft↗

arXiv2006

Interface mediated interactions between particles -- a geometrical approach

Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free surface stress tensor. This stress tensor can be used to express the force on a particle as a line integral along any conveniently chosen closed contour that surrounds the particle. The resulting expression is exact (i.e., free of any "smallness" assumptions) and independent of the chosen surface parametrization. Additional surface degrees of freedom, such as vector fields describing lipid tilt, are readily included in this formalism. As an illustration, we derive the exact force for several important surface Hamiltonians in various symmetric two-particle configurations in terms of the midplane geometry; its sign is evident in certain interesting limits. Specializing to the linear regime, where the shape can be analytically determined, these general expressions yield force-distance relations, several of which have originally been derived by using an energy based approach.

cond-mat.soft↗

arXiv2004

Geometry of surface mediated interactions

Soft interfaces can mediate interactions between particles bound to them. The force transmitted through the surface geometry on a particle may be expressed as a closed line integral of the surface stress tensor around that particle. This contour may be deformed to exploit the symmetries present; for two identical particles, one obtains an exact expression for the force between them in terms of the local surface geometry of their mid-plane; in the case of a fluid membrane the sign of the interaction is often evident. The approach, by construction, is adapted directly to the surface and is independent of its parameterization. Furthermore, it is applicable for arbitrarily large deformations; in particular, it remains valid beyond the linear small-gradient regime.

cond-mat.soft↗