Extracting shape from curvature evolution in moving surfaces

Abstract

Shape is a crucial geometric property of surfaces, interfaces, and membranes in biology, colloidal and interface science, and many areas of physics. This paper presents theory, simulation and scaling of local shape and curvedness changes in moving surfaces and interfaces, under uniform normal motion, as in phase ordering transitions in liquid crystals. Previously presented measures of shape and curvedness are introduced in quantities and equations used in colloidal science and interfacial transport phenomena to separate shape effects from those of curvedness. Considering in parallel the new shape formalism with the classical curvature formalism, this paper sheds new light on what effects originate only from shape. The new shape evolution equations are solved under uniform normal surface flow. It is found that the solutions obey the so-called "astigmatism equation" fixing the linear relation between the radii of curvature. Astigmatic trajectories in the shape-curvedness phase plane, can be clearly classified into two modes: (i) constant shape evolution, and (ii) variable shape-variable curvedness. Shapes between spheres and cylinders follow the former mode for large curvedness and transition at smaller curvedness into the latter. Shapes' transitions between cylinder and saddles only follow the second mode. Under geometry-driven stagnation (i.e. zero normal velocity) shapes can be frozen. Evolving spheres and cylinders freeze into the same original shape, but perturbed cylinders can freeze into a variety of shapes including saddles. The results provide a useful complementary view on how to describe and control shape evolution in surfaces and interfaces, of wide interest in soft matter materials.

Rate of Entropy Production in Evolving Interfaces and Membranes under Astigmatic Kinematics: Shape Evolution in Geometric-Dissipation Landscapes

“…The shape of surfaces, interfaces and membranes can be efficiently described by a normalized scalar dimensionless shape parameter ( S ) that discriminates between patches of spheres, cylinders, saddles and intermediate states [ 1 ]. Likewise, surface curvedness ( C ) or deviation from planarity, can be characterized by a positive scalar (the condition reduces to a plane if ), with natural units of inverse length.…”

“…An important aspect in equilibrium and non-equilibrium self-assembly, growth, phase ordering transitions, phase separation, accretion, abrasion and morphogenesis is the spatio-temporal evolution of shape and curvedness as controlled by kinematics and dynamics [ 33 ]. Since interfacial and membrane dissipation during evolving shape and curvedness [ 1 ] involves rates of change of bending and torsion, there is a natural and direct connection between entropy production rates and time-dependent geometric variations. The connection between geometry and thermodynamics has been long studied mainly in equilibrium and statics.…”

“…In this work, we initiate a similar framework but for dissipative evolution and growth, seeking to establish how the rate of entropy production manifests in shape and curvedness temporal changes, say, from a highly curved cylindrical patch to a flatter saddle patch. To maintain a realistic scope for the paper, we only consider surface patches, as opposed to closed shapes, and investigate viscous dissipation in moving and deforming surfaces, interfaces and membranes under given constant growth kinematics, where the velocity has only a normal component hence the surface curvatures are simply related to each other, a condition known as astigmatic flow since it corresponds to surfaces whose time-dependent curvatures are related to each other by a constant [ 1 ]. Other surface kinematics, such as mean curvature flow or Ricci flow are not treated in this work.…”

“…Figure 1 shows a schematic that summarizes the key objectives, the technical concepts, workflow, and the nomenclature used in the paper. To achieve the characterization of rates of entropy production due to evolving physical surface geometry under astigmatic flow we perform the following workflow steps: Using the well-known Boussinesq-Scriven dissipation [ 1 , 42 , 43 ] due to bending and torsion rates, we reformulate this quantity in terms of curvedness and shape. This is done by expressing bending and twisting rates in terms of actual curvedness and shape and not their rates.…”

“…We note that the line is a flat surface with no shape. The nomenclature in Figure 1 b which distinguish the sign of the shape are more commonly used in engineering; As this work only considers astigmatic flow [ 1 ], we establish and study the evolution lines in detail, given by the astigmatic flow: , where m is an invariant that defines a particular shape evolution trajectory. The evolutions are planar but curved lines on the thermodynamic surface under the -frame.…”