Curvature strains

Figure 7 Schematic models for the carbonaceous mesophase [after Mochida et al. (16)] and its curvature strains of bend, splay, and twi.st. The molecular and strain models differ in scale by 1000x.

P SS NMR data of 14-mer amphipathic peptide (composed of leucines and phenylalanines modified by crown ethers) embedded into membranes were reported by Ouellet et al.125 The preliminary study indicated that the 14-mer peptide remains at the surface of bilayers and perturbs the lipid orientation relative to the magnetic field. 15N-31P REDOR experiments have also been used to measure the intermolecular dipole-dipole interaction between the 14-mer peptide and the PL headgroup and polar region of DMPC MLVs. The results strongly suggest that the 14-mer peptide destabilises the lipid bilayer via the induction of a positive curvature strain. 

The lipid bilayer is not passive in determining membrane protein activity and function, and an accumulating body of evidence indicates that there is a coupling of membrane proteins to lipid bilayer properties. These properties include the effect of bilayer curvature strain (20), the role of specific lipids such as phosphoinositides, (21) and the effect of thickness on membrane protein function (22). The lipid composition, as well as the bilayer properties that result from this composition, act as allosteric regulators of membrane protein function. 

Let us take note that the molecular interactions giving rise to liquid crystal properties must also be present in solids. This indicates that conventional elastic theory is incomplete the direct curvature-strain moduli should also be included. This is true, but does not seriously invalidate the accepted theory of elasticity. Consider the bending of a beam, of thickness 2a, to a radius R. TThen the stored free energy according to ordinary elasticity theory is g — EayiAR, where E... 

Figure 11. A conneclional technique for narrowing, and introducing five- and seven-metnbered ring pairs to the graphitic tube of a torus in order to alleviate curvature strain. (From Ref. 26.)...

Next there are some inequalities that apply to the absolute curvatures appearing in the curvature-strain expression of Section 9.3.2. These allow often fairly tight bounds on the corresponding strains. First we have ... 

Much as already noted in Section 9.2.2 for finite cycles one could imagine knotting or linking for the infinite chain case. These would, however, generally entail excess curvature strain. 

Relationship between elasticity and orientational order As remarked in chapter I, a uniformly oriented film of nematic liquid crystal may be prepared by prior treatment of the surfaces with which it is in contact. If the preferred orientation imposed by the surfaces is perturbed, let us say by a magnetic field, a curvature strain will be introduced in the medium. The theory of such a deformation will be discussed at length in 3.2 for the present it will suffice to state some of the important results. The free energy per unit volume of the deformed medium relative to the state of uniform orientation is... 

We neglect higher order terms in the expansion as we are concerned only with infinitesimal deformations. Since these deformations relate to changes in the orientation of the director, may appropriately be called the curvature strain tensor (Frank ). In order to define the components of this tensor, let us choose a local right-handed system of cartesian coordinates with n parallel to z at the origin. The components of strain are then... 

We ignore dnjdx, nj y and dnjdz. The three types of deformation are shown in fig. 2.3.12. The curvature strain tensor may therefore be written as... 

A recent survey [70] classifies the vibration-based damage identification methods into four categories (1) natural frequency-based methods (2) mode shape-based methods (3) curvature/strain mode shape-based methods (4) other methods based... 

The curvature/strain modes method hold the promise of detecting damage from the processing of experimental data without a structural model. The premise of the method is that damage in thin-wall structures induces local discontinuities that affect strongly the curvature of the flexural modes of the structure [71]. The curvamre of the modes (i.e. the curvature modes) can be determined through space-wise double differentiation of the measured displacement/velocity/acceleration modes. They can be also measured directly with surface mounted strain gauges. 

Figure 3. Schematic of director curvature strains splay, twist, and bend (after Stephen and Straley).

On the basis of research carried out it was concluded that oxidation at lower temperature takes place due to higher curvature and associated lattice strain (Joshi et al. 1990 Yao et al. 1998 Singh et al. 2010). Bond curvature has been found to affect the oxidation of single-walled CNTs (Miyata et al. 2007). Therefore, nanotubes with smaller diameters, due to higher curvature strain, are oxidized at lower temperature. Defects and derivatization moiety in nanotube walls can also lower the thermal stability (Arepalli et al. 2004). Therefore, a higher oxidation temperature is always associated with purer, less defective samples. 

The only curvature strains of the director field which must be considered correspond to the splay, bend, and twist distortions (Fig. 2.17). Other types of deformation either do not change the elastic energy (e.g., the above mentioned pure shears) or are forbidden due to the symmetry. In nematic liquid crystals the cylindrical symmetry of the structure, as well as the absence of polarity (head to tail symmetry) must be taken into account. 

Deformation of the director field away from its equilibrium configuration increases the free-energy density of the mesophase. The curvature strains (and associated... 

Fig. 5.16. The director-field patterns in volume elements subjected to splay, twist, and bend curvature strains.

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