曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

DIRECT MANIPULATION OF B-SPLINE SURFACES

Engineering design and geometric modeling often require the ability to modifythe shape of parametric curves and surfaces so that their shape satisfy some given geometricconstraints, including point, normal vector, curve and surface. Two approaches are presented todirectly manipulate the shape of B-spline surface. The former is based on the least-square, whereasthe latter is based on minimizing the bending energy of surface. For each method, since unified andexplicit formulae are derived to compute new control points of modified surface, these methods aresimple, fast and applicable for CAD systems. Algebraic technique is used to simplify the computationof B-spline composition and multiplication. Comparisons and examples are also given.

Pseudo-break imaging of carbon nanotubes for determining elastic bending energies

One-dimensional(1D)nanomaterials easily bend due to perturbations from their surroundings or their own behaviors.This phenomenon not only impacts the performances of various devices but has also been employed to develop a variety of new functional devices,in which the bending energies of the nanomaterials determine the device performances.However,measuring the energies of such nanomaterials is extremely difficult.Herein,pseudo-break imaging of 1D nanomaterials has been proposed and realized on individual carbon nanotubes(CNTs),in which a CNT appears to break and has a fracture but is actually intact.This imaging approach provides the values of the bending energies of the CNTs with an accuracy of 1–50 eV.Furthermore,this imaging approach can manipulate the bending shapes and energies of CNTs.This work presents a protocol for bending analysis and manipulation,which are vital to fundamental and applied studies of 1D nanomaterials.

ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR A PHASE FIELD BENDING ELASTICITY MODEL OF VESICLE MEMBRANE DEFORMATION

In this paper, we study numerical approximations of a recently proposed phase field model for the vesicle membrane deformation governed by the variation of the elastic bending energy. To overcome the challenges of high order nonlinear differential systems and the nonlinear constraints associated with the problem, we present the phase field bending elasticity model in a nested saddle point formulation. A mixed finite element method is then employed to compute the equilibrium configuration of a vesicle membrane with prescribed volume and surface area. Coupling the approximation results for a related linearized problem and the general theory of Brezzi-Rappaz-Raviart, optimal order error estimates for the finite element approximations of the phase field model are obtained. Numerical results are provided to substantiate the derived estimates.

Construction of multi-homojunction TiO_(2)nanotubes for boosting photocatalytic hydrogen evolution by steering photogenerated charge transfer

As an effective means to improve charge carrier separation efficiency and directional transport,the gradient doping of foreign elements to build multi-homojunction structures inside catalysts has received wide attentions.Herein,we reported a simple and robust method to construct multi-homojunctions in black TiO_(2) nanotubes by the gradient doping of Ni species through the diffusion of deposited Ni element on the top of black TiO2 nanotubes driven by a high temperature annealing process.The gradient Ni distribution created parts of different Fermi energy levels and energy band structures within the same black TiO_(2) nanotube,which subsequently formed two series of multi-homojunctions within it.This special multi-homojunction structure largely enhanced the charge carrier separation and transportation,while the low concentration of defect states near the surface layer further inhibited carrier recombination and facilitated the surface reaction.Thus,the B-TNT-2Ni sample with the optimized Ni doping concentration exhibited an enhanced hydrogen evolution rate of~1.84 mmol·g^(−1)·h^(−1)under visible light irradiation without the assistance of noble-metal cocatalysts,~four times higher than that of the pristine black TiO_(2)nanotube array.With the capability to create multi-homojunction structures,this approach could be readily applied to various dopant systems and catalyst materials for a broad range of technical applications.