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We study thin films of A/B diblock copolymer in a lamellar phase confined between two parallel plates (electrodes) that impose a constant voltage across the film. The weak-segregation limit is explored via a Ginzburg-Landau-like free-energy expansion. We focus on the relative stability of parallel and perpendicular orientations of the lamellar phase, and how they are affected by variation of four experimental controllable parameters: addition of free ions, the difference in ionic solubilities between the A and B blocks, the dielectric contrast between the A/B blocks, and the preferential interaction energy of the plates with the blocks. It is found that, in general, the addition of ions lowers the critical voltage needed to reorient the lamellae from being parallel to the plates, to being perpendicular to them. The largest reduction in critical voltage is obtained when the ions are preferentially solubilized in the block that is not preferred by the plates. This reduction is even further enhanced when the dielectric constant of this block has the higher value. These predictions are all subject to experimental verification., Comment: 14 pages, 8 figures

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 figures

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 figures

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, 2005

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 purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed lipid bilayers, the shape equation and boundary conditions of open lipid bilayers and two-component membranes, the shape equation and in-plane strain equations of cell membranes with cross-linking structures, and the stabilities of closed lipid bilayers and cell membranes. The key point of this scheme is to deal with the variational problems on the surfaces embedded in three-dimensional Euclidean space by using exterior differential forms., Comment: 25 pages, 1 figure

The entropy of a polymer confined in a curved surface and the elastic free energy of a membrane consisting of polymers are obtained by scaling analysis. It is found that the elastic free energy of the membrane has the form of the in-plane strain energy plus Helfrich's curvature energy [Z. Naturforsch. C \textbf{28}, 693 (1973)]. The elastic constants in the free energy are obtained by discussing two simplified models: one is the polymer membrane without in-plane strains and asymmetry between its two sides, which is the counterpart of quantum mechanics in curved surface [Jensen and Koppe, Ann. Phys. \textbf{63}, 586 (1971)]; another is the planar rubber membrane with homogeneous in-plane strains. The equations to describe equilibrium shape and in-plane strains of the polymer vesicles by osmotic pressure are derived 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 critical pressure, above which spherical polymer vesicle will lose its stability, is obtained by taking the second order variation of the total free energy. It is found that the in-plane mode also plays important role in the critical pressure because it couples with the out-of-plane mode. Theoretical results reveal that polymer vesicles possess the mechanical properties intermediate between fluid membranes and solid shells., Comment: 17 pages, 1 figure, 1 table

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 figures

An elegant formula for coordinates of carbon atoms in a unit cell of a single-walled nanotube (SWNT) is presented and the potential of neon (Ne) inside an infinitely long SWNT is analytically derived out under the condition of the Lennard-Jones potential between Ne and carbon atoms. Specific heats of dilute Ne inside long (20, 20) SWNT are calculated at different temperatures. It is found that Ne exhibits 3-dimensional (3D) gas behavior at high temperature but behaves as 2D gas at low temperature. Especially, at ultra low temperature, Ne inside (20, 20) nanotubes behaves as lattice gas. A coarse method to determine the characteristic temperature $\mathcal{T}_c$ for low density gas in a potential is put forward. If $\mathcal{T}\gg \mathcal{T}_c$, we just need to use the classical statistical mechanics without solving the Shr\"{o}dinger equation to consider the thermal behavior of gas in the potential. But if $\mathcal{T}\sim \mathcal{T}_c$, we must solve the Shr\"{o}dinger equation. For Ne in (20,20) nanotube, we obtain $\mathcal{T}_c\approx 60$ K., Comment: 14 pages, 7 figures

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 PRB

An elegant formula for coordinates of carbon atoms in a unit cell of a single-walled nanotube (SWNT) is presented and a new molecular motor of double-walled carbon nanotube whose inner tube is a long (8,4) SWNT and outer tube a short (14,8) SWNT is constructed. The interaction between inner an outer tubes is analytically derived by summing the Lennard-Jones potentials between atoms in inner and outer tubes. It is proved that the molecular motor in a thermal bath exhibits a directional motion with the temperature variation of the bath., Comment: 9 pages, 4 figures, revtex4

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

A simple formula for coordinates of carbon atoms in a unit cell of a single-walled nanotube (SWNT) is presented and the potential of neon (Ne) inside an infinitely long SWNT is analytically derived under the assumption of pair-wise Lennard-Jones potential between Ne and carbon atoms. Specific heats of dilute Ne inside infinitely long (5, 5), (10, 10), (15, 15) and (20, 20) SWNT's are calculated at different temperatures. It is found that Ne inside four kinds of nanotubes exhibits 3-dimensional (3D) gas behavior at high temperature but different behaviors at low temperature: Ne inside (5, 5) nanotube behaves as 1D gas but inside (10, 10), (15, 15), and (20, 20) nanotubes behaves as 2D gas. Furthermore, at ultra low temperature, Ne inside (5, 5) nanotube still displays 1D behavior but inside (10, 10), (15, 15), and (20, 20) nanotubes behaves as lattice gas., Comment: 10 pages, 5 figures

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)