Cylindrical curvature

In a cylindrical pore the meniscus will be spherical in form, so that the two radii of curvature are equal to one another and therefore to r (Equation (3.8)). From simple geometry (Fig. 3.8) the radius r of the core is related to r by the equation... 

Textile fibers must be flexible to be useful. The flexural rigidity or stiffness of a fiber is defined as the couple required to bend the fiber to unit curvature (3). The stiffness of an ideal cylindrical rod is proportional to the square of the linear density. Because the linear density is proportional to the square of the diameter, stiffness increases in proportion to the fourth power of the filament diameter. In addition, the shape of the filament cross-section must be considered also. For textile purposes and when flexibiUty is requisite, shear and torsional stresses are relatively minor factors compared to tensile stresses. Techniques for measuring flexural rigidity of fibers have been given in the Hterature (67—73). 

Vertical pre.s.sure leaf filters. These filters have vertical, paraUel, rec tangular leaves mounted in an upright cylindrical pressure tank. The leaves usually are of such different widths as to aUowthem to conform to the curvature of the tank and to fill it without waste space. The leaves often rest on a filtrate manifold, the connec tion being sealed by an O ring, so that they can be lifted individuaUy from the top of the fil-... 

The first observation is that the cured shape of an unsymmetric cross-ply laminate is often cylindrical, whereas we would predict it to be a saddle shape (hyperbolic paraboloid) from classical lamination theory (the curvatures can be shown to be = - Ky or - = Ky). A thick lami-... 

For a thicker laminate than in Figure 6-26, the critical length is longer and the curvatures are smaller. For example, for a [04/904]-,-laminate, the critical L is 71 mm. Moreover, what was a circular cylindrical specimen at 50 mm for a [02/902lx laminate becomes a saddle-shaped specimen [6-38]. 

It is known that a metallic ID system is unstable against lattice distortion and turns into an insulator. In CNTs instabilities associated two kinds of distortions are possible, in-plane and out-of-plane distortions as shown in Fig. 8. The inplane or Kekuld distortion has the form that the hexagon network has alternating short and long bonds (-u and 2u, respectively) like in the classical benzene molecule [8,9,10]. Due to the distortion the first Brillouin zone reduees to one-third of the original one and both K and K points are folded onto the F point in a new Brillouin zone. For an out-of-plane distortion the sites A and B are displaced up and down ( 2) with respect to the cylindrical surface [11]. Because of a finite curvature of a CNT the mirror symmetry about its surface are broken and thus the energy of sites A and B shift in the opposite direction. 

The synthesis of molecular carbon structures in the form of C q and other fullerenes stimulated an intense interest in mesoscopic carbon structures. In this respect, the discovery of carbon nanotubes (CNTs) [1] in the deposit of an arc discharge was a major break through. In the early days, many theoretical efforts have focused on the electronic properties of these novel quasi-one-dimensional structures [2-5]. Like graphite, these mesoscopic systems are essentially sp2 bonded. However, the curvature and the cylindrical symmetry cause important modifications compared with planar graphite. 

Baumgartner and coworkers [145,146] study lipid-protein interactions in lipid bilayers. The lipids are modeled as chains of hard spheres with heads tethered to two virtual surfaces, representing the two sides of the bilayer. Within this model, Baumgartner [145] has investigated the influence of membrane curvature on the conformations of a long embedded chain (a protein ). He predicts that the protein spontaneously localizes on the inner side of the membrane, due to the larger fluctuations of lipid density there. Sintes and Baumgartner [146] have calculated the lipid-mediated interactions between cylindrical inclusions ( proteins ). Apart from the... 

J. Holuigue, O. Bertrand, E. Arquis. Solutal convection in crystal growth effect of interface curvature on flow structuration in a three-dimensional cylindrical configuration. J Cryst Growth 180 591, 1997. 

For a cylindrical vessel, the radius of curvature in the axial direction is infinite, and the stress in the direction of the circumference, called the hoop stress, is... 

The radius of curvature r of the meniscus and its depth H are expressed for a cylindrical micro-channel as... 

The surface forces apparatus (SEA) can measure the interaction forces between two surfaces through a liquid [10,11]. The SEA consists of two curved, molecularly smooth mica surfaces made from sheets with a thickness of a few micrometers. These sheets are glued to quartz cylindrical lenses ( 10-mm radius of curvature) and mounted with then-axes perpendicular to each other. The distance is measured by a Fabry-Perot optical technique using multiple beam interference fringes. The distance resolution is 1-2 A and the force sensitivity is about 10 nN. With the SEA many fundamental interactions between surfaces in aqueous solutions and nonaqueous liquids have been identified and quantified. These include the van der Waals and electrostatic double-layer forces, oscillatory forces, repulsive hydration forces, attractive hydrophobic forces, steric interactions involving polymeric systems, and capillary and adhesion forces. Although cleaved mica is the most commonly used substrate material in the SEA, it can also be coated with thin films of materials with different chemical and physical properties [12]. 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

Cylindrical sections are usually made up from plate sections rolled to the required curvature. The sections (strakes) are made as large as is practicable to reduce the number of welds required. The longitudinal welded seams are offset to avoid a conjunction of welds at the comers of the plates. 

Fig. 60 Schematic illustration for formation of cylindrical morphology in a blend of slightly asymmetric lower molecular weight PS-b-PI (/3-chain) with large symmetric PS-fc-PI (a-chain). a Molecule of /S-chain with non-zero spontaneous curvature, b Cylindrical morphology formed by neat /3 chains shown in a. Here mean curvature of cylinder (solid line) is larger than spontaneous curvature of /3-chain (dashed lines). c Cylindrical morphology formed by binary blend of /3-chains shown in a and large symmetric copolymers (a-chain). In this case, mean curvature of cylinder closely fits to spontaneous curvature of /3-chain. From [180]. Copyright 2001 American Chemical Society...

It has been known that the basal graphite plane (graphene hexagon) is chemically inert. However, CNTs are susceptive to some chemical reactions due to the it-orbital mismatch in the curvature structures. Oxidation studies have revealed that the tips (caps) of CNTs are more reactive than the cylindrical parts [8, 20], Ab initio calculations indicate that the average charge density of a pentagon (at the tips) is 3 4 times larger... 

K being the leaf-spring constant) and by starting measurements with the surfaces far apart, where F(D) = 0, force profiles may be determined. The method also allows the measurement of the mean refractive index n(D) of the medium between the surfaces (9), and from this the amount r of adsorbed species per unit area of mica may be evaluated. Finally the mean radius of curvature R of the cylindrically curved mica surfaces near the contact area may also be calculated from the shape of the interference fringes. 

The three-dimensional APH display may be conveniently chosen to map onto a simple cylindrical surface (e.g., a coffee cup) by adding blank filling space to each helical turn in order to maintain a constant radius of curvature. Figure B.3 provides a simple planar template that one can cut and paste to form the APH cylindrical display. 

Furthermore, Oda et al. pointed out that there are two topologically distinct types of chiral bilayers, as shown in Figure 5.46.165 Helical ribbons (helix A) have cylindrical curvature with an inner face and an outer face and are the precursors of tubules. These are, for example, the same structures that are observed in the diacetylenic lipid systems discussed in Section 4.1. By contrast, twisted ribbons (helix B) have Gaussian saddlelike curvature, with two equally curved faces and a C2 symmetry axis. They are similar to the aldonamide and peptide ribbons discussed in Sections 2 and 3, respectively. The twisted ribbons in the tartrate-gemini surfactant system were found to be stable in water for alkyl chains with 14-16 carbons. Only micelles form... 

Figure 5.46 Schematic representation of helical and twisted ribbons as discussed in Ref. 165. Top Platelet or flat ribbon. Helical ribbons (helix A), precursors of tubules, feature inner and outer faces. Twisted ribbons (helix B), formed by some gemini surfactant tartrate complexes, have equally curved faces and C2 symmetry axis. Bottom Consequences of cylindrical and saddlelike curvatures in multilayered structures. In stack of cylindrical sheets, contact area from one layer to next varies. This is not the case for saddlelike curvature, which is thus favored when the layers are coordinated. Reprinted with permission from Ref. 165. Copyright 1999 by Macmillan Magazines.

The experiments discussed in this chapter have shown that a variety of chiral molecules self-assemble into cylindrical tubules and helical ribbons. These are indeed surprising structures because of their high curvature. One would normally expect the lowest energy state of a bilayer membrane to be flat or to have the minimum curvature needed to close off the edges of the membrane. By contrast, these structures have a high curvature, with a characteristic radius that depends on the material but is always fairly small compared with vesicles or other membrane structures. Thus, the key issue in understanding the formation of tubules and helical ribbons is how to explain the morphology with a characteristic radius. 

In this chapter, we have surveyed a wide range of chiral molecules that self-assemble into helical structures. The molecules include aldonamides, cere-brosides, amino acid amphiphiles, peptides, phospholipids, gemini surfactants, and biological and synthetic biles. In all of these systems, researchers observe helical ribbons and tubules, often with helical markings. In certain cases, researchers also observe twisted ribbons, which are variations on helical ribbons with Gaussian rather than cylindrical curvature. These structures have a large-scale helicity which manifests the chirality of the constituent molecules. 

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