Fold period

Zatsepin V. M. Time correlation functions of one-dimensional rotational Brownian motion in n-fold periodical potential. Theor. and Math. Phys. 

Chen et ai, 2002), which exhibits eight-fold periodicity along the [110] azimuth and incommensurate order along [001] and HBC/Au(lll) (Sellam et ai, 2001) -3 6 ... 

Proper description of the torsional potential requires a form that reflects its inherent periodicity. For example, the three-fold periodicity of rotation about the carbon-carbon bond in ethane may be described by the simple functional form. 

CO is the torsion angle, co i is the ideal torsion angle and is treated as a parameter. Bond torsion contributions to the overall energy may also need to include terms which are one-fold and two-fold periodic. 

While this well defined band structure was originally interpreted as indicating a very narrow molecular-weight distribution, it now seems more probable to the present authors that the chains are folded as they are in other crystalline polymers. The long fold period may well be related to the great stiffness of the fluorocarbon chains. 

McLachlan, A. D., and Stewart, M. (1976). The 14-fold periodicity in alpha-tropomyosin and the interaction with actin./. Mol. Biol. 103, 271-298. 

Details of the elemental and neutron periodicities follow directly as subsets of the 24-fold periodic function of the nucleons. The periodic ordinal numbers, derived in this way, define the stability limits of nuclides in terms of, either atomic number or neutron number, as shown in Figures 4.4 and 4.11 respectively. 

Note the inverse dependence of fold period on AT. The larger the undercooling the smaller the fold period. 

Polymer crystal growth is predominantly in the lateral direction, because folds and surface entanglements inhibit crystalliza- 4 don in the thickness direction. Neverthe-1 less, there is a considerable increase in the fold period behind the lamellar front during crystallization from the melt and, as we have j seen, polymers annealed above their crys-tallization temperature but below Tm also irreversibly thicken. Nevertheless, in most theories of secondary nucleation, the most i widely used being the theory of Lauritzen and Hoffman,28 it is assumed that once a part of a chain is added to the growing crystal, its. fold period remains unchanged. 

In this theory it is assumed that a chain stem, one fold period long, is laid down on the lateral growth face of the crystal. This is the slowest step because the stem has only one surface face on which to sit. Once this stem is in place, however, an adjacent stem is more easily laid down (i.e., has a lower free energy barrier to cross), because it can now contact two surfaces, the crystal substrate and the side face of the first stem. The row therefore quickly fills up once the first stem is deposited. The growth rate of the crystal (primary crystallization) is thus largely determined by secondary nucleation (Figure 10-31). 

This is half the value determined for primary nucleation. But, as in primary nucle-ation, the fold period goes as 1/tsT. In other words, it increases with decreasing undercooling (smaller AT). The temperature dependence of the rate of secondary nucleation has a different dependence on temperature, however. 

So, to summarize, the theory predicts that the fold period (thickness) of the crystals increases with decreasing undercooling (i.e., higher temperature), while the rate of primary crystallization increases with decreasing temperature, at least at low undercoolings (more on this in a bit). 

FIGURE 10-33 The fold period of PE versus temperature (redrawn, from the data of Hoffman et aL cited previously). 

FIGURE 10-42 Schematic graph of fold period versus temperature. 

The barrier to rotation around the C-C bond in ethane has three fold periodicity. Three conformations, in order of decreasing energy, are eclipsed, skewed and staggered. 

The rotational barrier in propane has three-fold periodicity. 

As rotation proceeds the energy of ethane therefore varies in a pattern that repeats itself three times per revolution. The energy profile associated with rotation about a C-C bond in propane likewise shows a three-fold periodicity. 

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