Twisting and Bending

A graded charge of rods results from wear in a rod mill. Rod diameter may range from 10 to 2.5 cm (4 to 1 in), for example. A new rod load usually is patterned after a used one found to give good results. The maximum length of a rod mill appears to be 20 rt, because longer rods tend to twist and bend. 

The first several chapters of any organic chemistry textbook focus on the structure of molecules how atoms connect to form bonds, how we draw those connections, the problems with our drawing methods, how we name molecules, what molecules look like in 3D, how molecules twist and bend in space, and so on. Only after gaining a clear understanding of stracture do we move on to reactions. But there seems to be one exception acid-base chemistry. 

The molecule is twisting and bending around all of the time, and the conformation with the bracelet-shaped skeleton is just one of the possible conformations. The molecule probably spends very little of its time like this (it is a relatively high energy conformation), but this is the conformation that we will use to draw our Fischer projection. 

Figure 4.3. The rod (or disk) model for torsion and flexure of DNA. The DNA is modeled as a string of rods (or disks) connected by Hookean twisting and bending springs which oppose, respectively, torsional and flexural deformations. The instantaneous z and x axes of a subunit rod around which the mean squared angular displacements , j = x, z, take place are indicated. The filament is assumed to exhibit mean local cylindrical symmetry in the sense that for any pair of transverse x- and y-axes. Twisting = mean squared angular displacement about body-fixed x -axis = (/)y(t)2) (assumed).

Yamakawa and co-workers have formulated a discrete helical wormlike chain model that is mechanically equivalent to that described above for twisting and bending/79111 117) However, their approach to determining the dynamics is very different. They do not utilize the mean local cylindrical symmetry to factorize the terms in r(t) into products of correlation functions for twisting, bending, and internal motions, as in Eq. (4.24). Instead, they... 

When the internal motion is so rapid that it relaxes before significant relaxation by twisting and bending takes place, there exists a time domain in which 7 (t)s/n(oo)yet C (t) = 1.0, F (t)s 1.0, and the residual anisotropy is given by... 

Barkley, M.D. and Zimm, B.H. (1979) Theory of twisting and bending of chain macromolecules analysis of the fluorescence depolarization of DNA. J. Chem. Phys. 70, 2991-3007. 

Note 1 The twist and bend distortions can be stabilised by an array of screw or edge dislocations. 

For a nematic LC, the preferred orientation is one in which the director is parallel everywhere. Other orientations have a free-energy distribution that depends on the elastic constants, K /. The orientational elastic constants K, K22 and K33 determine respectively splay, twist and bend deformations. Values of elastic constants in LCs are around 10 N so that free-energy difference between different orientations is of the order of 5 x 10 J m the same order of magnitude as surface energy. A thin layer of LC sandwiched between two aligned surfaces therefore adopts an orientation determined by the surfaces. This fact forms the basis of most electrooptical effects in LCs. Display devices based on LCs are discussed in Chapter 7. 

Chain flexibility is explicitly acknowledged, allowing some chains to twist and bend in such a way as to fill wedges that would otherwise contain water. 

Fig. 5. Various parameters of accessibility, twist, and bend plotted vs. sequence number. Part 1 (a) Solvent-accessible area of side chains, (b) Fractional accessibility (referred to full sphere) of backbone carbonyl oxygen and peptide nitrogen. The separate plot for values less than 1% is meant to show that no accessibility was detected for many atoms. The actual nonzero values are not to be taken too literally. Part 2 (c) Backbone angles as normally defined, (d) Angles between sequentially adjacent carbonyl vectors in the backbone plotted between the sequence numbers of the two residues involved. Part 3 (e) Distance in A between the tips, T, of adjacent residues as defined in the text, (f) Distances in A between peptide center, M, and the third sequential peptide center (open circles), and between carbon a and the sixth sequential a-carbon (crosses) plotted opposite the central carbon atom in each case, (g) Angles between lines joining the centers of successive peptide bonds plotted between the residues defining the central bond, (h) Angles between lines joining successive a carbons plotted opposite the central carbon, (Note that the accessibilities were calculated with coordinate set 4 and the other parameters with set 6 see text.)...

Wang X, Zho J-J, Li L (2004) Twisting and bending of polyethylene crystals. Paper presented at PP 2004, Dali (June)... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

More complicated molecules twist and bend to make sure that all the atoms take up positions that allow them maximum free space and non-interference with each other. The very complicated DNA molecules with hundreds of atoms in them twist in a characteristic spiral manner (Figure 2.10). The characteristic shape of each molecule influences its effect on how it behaves in our body and within cells. 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

Figure 2.9 Schematic representation of the elastic constants for splay, twist and bend, ku, k22 und kjs, respectively, of a nematic phase.

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