Structures, Symmetries, Mechanics and Motors of carbon nanotubes

The structures and symmetries of single-walled carbon nanotubes (SWNTs) are introduced in detail. The physical properties of SWNTs induced by their symmetries can be described by tensors in mathematical point of view. It is found that there are 2, 4, and 5 different parameters in the second, third, and fourth rank tensors representing electronic conductivity (or static polarizability), the second order nonlinear polarizability, and elastic constants of SWNTs, respectively. The values of elastic constants obtained from tight-binding method imply that SWNTs might be very weakly anisotropic in mechanical properties. The further study on the mechanical properties shows that the elastic shell theory in the macroscopic scale can be applied to carbon nanotubes (CNTs) in the mesoscopic scale, as a result, SWNTs can be regarded as an isotropic material with Poisson ratio, effective thickness, and Young's modulus being $\nu=0.34$, $h=0.75$\AA, $Y=4.70$TPa, respectively, while the Young's moduli of multi-walled carbon nanotubes (MWNTs) are apparent functions of the number of layers, $N$, varying from 4.70TPa to 1.04TPa for N=1 to $\infty$. Based on the chirality of CNTs, it is predicted that a new kind of molecular motor driven by alternating voltage can be constructed from double walled carbon nanotubes (DWNTs).Comment: 18 pages+5 figure; will appear as a Chapter in "Nanotubes: New Research" (Nova Science Publishers, 2005

Lipid membranes with free edges

Lipid membrane with freely exposed edge is regarded as smooth surface with curved boundary. Exterior differential forms are introduced to describe the surface and the boundary curve. The total free energy is defined as the sum of Helfrich's free energy and the surface and line tension energy. The equilibrium equation and boundary conditions of the membrane are derived by taking the variation of the total free energy. These equations can also be applied to the membrane with several freely exposed edges. Analytical and numerical solutions to these equations are obtained under the axisymmetric condition. The numerical results can be used to explain recent experimental results obtained by Saitoh \emph{et al}. [Proc. Natl. Acad. Sci. \textbf{95}, 1026 (1998)].Comment: 15 pages, 6 figure

Comment on "Highly Extended Image States around Nanotubes"

A Comment on the Letter by Granger et.al., Phys. Rev. Lett. 89, 135506 (2002).Comment: 2 page

Recent theoretical advances in elasticity of membranes following Helfrich's spontaneous curvature model

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solutions to these equations and their corresponding configurations are demonstrated.Comment: 10 pages, 8 figure

Double-walled carbon nanotubes as hundred gigahertz oscillators

Based on the van der Waals interaction, the periodically nonlinear potential of a singe-walled carbon nanotube (SWNT) with finite length in an infinite length SWNT is analytically obtained. It is found that the inner SWNT can oscillate in the outer SWNT with frequency beyond ten Gigahertz, even up to a hundred Gigahertz.Comment: 9 pages, 8 figures, to PR

Variational Problems in Elastic Theory of Biomembranes, Smectic-a Liquid Crystals, and Carbon Related Structures

After a brief introduction to several variational problems in the study of shapes of thin thickness structures, we deal with variational problems on 2-dimensional surface in 3-dimensional Euclidian space by using exterior differential forms. The morphological problems of lipid bilayers and stabilities of cell membranes are also discussed. The key point is that the first and the second order variations of the free energy determine equilibrium shapes and mechanical stabilities of structures.Comment: 12 pages + 3 figures. For the Seventh International Conference on Geometry, Integrability and Quantization, Varna, 200

Elastic theory of low-dimensional continua and its applications in bio- and nano-structures

This review presents the elastic theory of low-dimensional (one- and two-dimensional) continua and its applications in bio- and nano-structures. First, the curve and surface theory, as the geometric representation of the low-dimensional continua, is briefly described through Cartan moving frame method. The elastic theory of Kirchhoff rod, Helfrich rod, bending-soften rod, fluid membrane, and solid shell is revisited. Secondly, the application and availability of the elastic theory of low-dimensional continua in bio-structures, including short DNA rings, lipid membranes, and cell membranes, are discussed. The kink stability of short DNA rings is addressed by using the theory of Kirchhoff rod, Helfrich rod, and bending-soften rod. The lipid membranes obey the theory of fluid membrane. A cell membrane is simplified as a composite shell of lipid bilayer and membrane skeleton, which is a little similar to the solid shell. It is found that the membrane skeleton enhances highly the mechanical stability of cell membranes. Thirdly, the application and availability of the elastic theory of low-dimensional continua in nano-structures, including graphene and carbon nanotubes, are discussed. A revised Lenosky lattice model is proposed based on the local density approximation. Its continuum form up to the second order terms of curvatures and strains is the same as the free energy of 2D solid shells. Several typical mechanical properties of carbon nanotubes are revisited and investigated based on this continuum form. It is possible to avoid introducing the controversial concepts, the Young's modulus and thickness of graphene and single-walled carbon nanotubes, with this continuum form.Comment: Review article for J. Comput. Theor. Nanosci., 27 pages, 15 figure

Elasticities and stabilities: lipid membranes vs cell membranes

A cell membrane can be simply regarded as composite material consisting of lipid bilayer, membrane cytoskeleton beneath lipid bilayer, and proteins embedded in lipid bilayer and linked with membrane cytoskeleton if one only concerns its mechanical properties. In this Chapter, above all, the authors give a brief introduction to some important work on mechanical properties of lipid bilayers following Helfrich's seminal work on spontaneous curvature energy of lipid bilayers. Next, the entropy of a polymer confined in a curved surface and the free energy of membrane cytoskeleton are obtained by scaling analysis. It is found that the free energy of cell membranes has the form of the in-plane strain energy plus Helfrich's curvature energy. The equations to describe equilibrium shapes and in-plane strains of cell membranes by osmotic pressures are obtained by taking the first order variation of the total free energy containing the elastic free energy, the surface tension energy and the term induced by osmotic pressure. The stability of spherical cell membrane is discussed and the critical pressure is found to be much larger than that of spherical lipid bilayer without membrane cytoskeleton. Lastly, the authors try to extend the present static mechanical model of cell membranes to the cell structure dynamics by proposing a group of coupling equations involving tensegrity architecture of cytoskeleton, fluid dynamics of cytoplasm and elasticities of cell membranes.Comment: 14 pages. Accepted as a chapter in Soft-condensed matter: new researches (Nova, 2005

The q-nonadditivity of nonextensive statistics is not a true physical property

This is a note showing that, contrary to our lasting belief, the nonadditivity X(1+2)=X(1)+X(2)+\alpha X(1)X(2) is not a true physical property. \alpha in this expression cannot be unique for a given system. It unavoidably depends on how one mathematically divides the system and cannot be used to characterize nonadditivity. As a matter of fact, its use is mathematically inconsistent

Strong enhancement of the spin Hall effect by spin fluctuations near the Curie point of FexPt1-x alloys

Robust spin Hall effects (SHE) have recently been observed in non-magnetic heavy metal systems with strong spin-orbit interactions. These SHE are either attributed to an intrinsic band-structure effect or to extrinsic spin-dependent scattering from impurities, namely side-jump or skew scattering. Here we report on an extraordinarily strong spin Hall effect, attributable to spin fluctuations, in ferromagnetic FexPt1-x alloys near their Curie point, tunable with x. This results in a damping-like spin-orbit torque being exerted on an adjacent ferromagnetic layer that is strongly temperature dependent in this transition region, with a peak value that indicates a lower bound 0.34 (+-) 0.02 for the peak spin Hall ratio within the FePt. We also observe a pronounced peak in the effective spin-mixing conductance of the FM/FePt interface, and determine the spin diffusion length in these FexPt1-x alloys. These results establish new opportunities for fundamental studies of spin dynamics and transport in ferromagnetic systems with strong spin fluctuations, and a new pathway for efficiently generating strong spin currents for applications.Comment: 23 pages, 4 figures. Accepted in Physical Review Letter