Segment size

The very first question that comes to mind when dealing with giant telescopes is the cost-effective feasibility of its optics. Assuming classical materials for the segments blanks, however, there is no need for a very substantial increase in production capacity from existing suppliers provided that the segment size remains below 2-m. Moderately lightweight Silicon Carbide is also considered as a serious and potentially cost-effective candidate, for its superior thermal performance and specific stiffness. 

Another compelling reason to limit segment size is the cost of transportation beyond 2.3-m flat-to-flat segment size, transport in standard 40 ft containers is excluded and transport costs increase beyond reasonable limits. 

The two chains in the lipid bilayer are not identical, because they are positioned asymmetrically with respect to the PC head. It turns out that the tail closest to the head group is buried less deep in the bilayer than the other chain [98]. The difference is not very large it amounts to about half a segment size. From this difference, we can rationalise the disparities in order between the two tails. The chain that is pulled out will be stretched most, and the order tends to be higher than the other chain. Below we will see that the differences in the behaviour of the tails can become larger if there are differences between the tails, e.g. with respect to the degree of unsaturation. 

Figure 23. Radial segment density profile through a cross-section of a highly curved spherical vesicle. The origin is at r = 0, and the vesicle radius is very small, i.e. approximately r = 25 (in units of segment sizes). The head-group units, the hydrocarbons of the tails and the ends of the hydrocarbon tails are indicated. Calculations were done on a slightly more simplified system of DPPC molecules in the RIS scheme method (third-order Markov approximation), i.e. without the anisotropic field contributions...

Figure 9. Dependence of the scaled length SL on the projected segment size SS on a logarithmic scale obtained from the self-affine fractal profiles in Figure 7 by using the triangulation method for Euclidean two-dimensional space. Flere, S means d In SL / d In SS. Reprinted from H. -C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem. 531, p. 101, Copyright 2002, with permission from Elsevier Science.

The above operation is iterated at various segment sizes. Finally, the self-similar fractal dimension of the profile embedded by the two-dimensional space is given by... 

Most promising market segments, size, and gradient... 

Let us consider a slit-like pore of width D along whose walls the ip(x) potential is localized (Fig. 4). We shall regard the interaction of monomers with the walls as a short-range interaction and the characteristic radius of interaction as being of the order of the segment size a. The exact assignment of the form of the potential is immaterial for our purposes, since it describes the effective interaction of units with the pore walls, renormalized by the solvent molecules. Conditions are to be as follows ... 

Fig. 4a and b. Distribution of the segment density at different values of the energy 0 (a) and schematic picture of lattice-like chain of length N in a slit-like pore of width D (b). 0cis the critical energy characteristic of the case when the entropy losses of the macromolecule in the pore are compensated by the energy of interaction with the wall. attractive potential of a depth 0 and with a characteristic radius of interaction r0 of the order of the segment size a... 

Changing the depth of the potential 0, e.g. by changing the composition of the solvent or the temperature (since 0 is in kT units), one can find a certain value of 0c at which k = 0. We shall call such conditions critical. They correspond to the case when the entropy losses of the bond between two successive monomers close to the pore wall, at a distance of the order of the segment size a, are compensated by the interaction energy with the wall. 

To close this section we calculate the effective segment size... 

We fch.ua take the segment size smaller and. smaller, at the same time increasing the number of segments so as to keep the unperturbed coil size rv constant. Thus... 

The Edwards Hamiltonian is an appealing but most formal object. To mention a simple fact, shrinking to zero the segment size of the discrete chain model as done in the continuous chain limit, we in general get a continuous but not differentiable space curve. Strictly speaking the first part, of Vj, does not exist. Further serious mathematical problems are connected to the (5-function interaction. Hie question in which sense Ve is a mathematically well defined object beyond its formal perturbation expansion is ari interesting problem of mathematical physics. 

Clearly this problem is a microstmcture effect resulting from the continuous chain limit, which allows for self-interactions of arbitrarily small pieces of the chain. To avoid it, we introduce a cut off. We impose the rule that segment variables Sj are not allowed to approach each other along the chain on distances smellier than a > 0. In some sense this again introduces a finite segment size l a-1/2. Imposing the same cut off at the chain ends we thus write... 

In this case the mapping just expresses the relation among segment size and segment number which holds for the Gaussian chain. 

We should pause hen a little to put the argument in a somewhat larger perspective. As basic step we may consider the redefinition of the segment size -1- which can be viewed as a dilatation of the microscopic length... 

This shows that the variables of the renormalized theory depend on temperature, chain length, and segment size via the two parameters z, Ef. For T — 0, i.e. fa T — S - > 0, we find... 

---