Spherical segment

Hemoglobin and many enzymes are covalent polymers with a globular shape. This shape is enforced by the tendency of hydrophobic amino acids to form a hydrophobic droplet in aqueous solutions solubilized by hydrophilic side-chains around them. The same is true for synthetic block polymers made of hydrophobic and hydrophilic segments. " Spherical biopolymers thus usually appear as micelles, with a core made of organic material. Covalency allows the construction of fully organized micelles, e.g. dendrimeric spheres, where one half has a hydrophilic, the other a hydrophobic surface. Block polymers may not only form micelles, but they may also arrange to form vesicles which entrap a water volume. Such spheres have a thick polymer wall." Both the polymer micelles and vesicles can be removed from solution without collapsing. 

At the beginning of this section we enumerated four ways in which actual polymer molecules deviate from the model for perfectly flexible chains. The three sources of deviation which we have discussed so far all lead to the prediction of larger coil dimensions than would be the case for perfect flexibility. The fourth source of discrepancy, solvent interaction, can have either an expansion or a contraction effect on the coil dimensions. To see how this comes about, we consider enclosing the spherical domain occupied by the polymer molecule by a hypothetical boundary as indicated by the broken line in Fig. 1.9. Only a portion of this domain is actually occupied by chain segments, and the remaining sites are occupied by solvent molecules which we have assumed to be totally indifferent as far as coil dimensions are concerned. The region enclosed by this hypothetical boundary may be viewed as a solution, an we next consider the tendency of solvent molecules to cross in or out of the domain of the polymer molecule. 

Only a fraction of the chain segments will be present in this spherical shell, but whatever their number is, it will increase with the degree of polymerization n. Therefore, in the volume element associated with the expansion of the coil, the volume fraction of chain segments 0 is proportional to n/dV, or 0 n/a ro dro ... 

Dome-roof tanks are similar to umbrella-roof tanks except that the dome more neatly approximates a spherical surface than the segmented sections of an umbrella roof... 

Dished or Basket Heads consist of a spherical segment normally dished to a radius equal to the inside diameter of the tank cylinder (or within a range of 6 inches plus or minus) and connected to the straight cylindrical flange by a knuckle whose inside radius is usually not less than 6 per cent of the inside diameter of the cylinder nor less than 3 times the thickness of the head plate. Basket heads closely approximate hemi-ellipsoidal heads. 

Bumped Heads consist of a spherical segment joining the tank cylinder directly without the transition knuckle. The radius = D, or less. This type of head is used only for pressures of 10 pounds per square inch or less, excepting where a compression ring is placed at the junction of head and shell. 

The average thickness of the spherical grafted layer is determined by the requirement that the integral over the segment concentration profile must account for all the monomers in the layer ... 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

Morphology of the anionically synthesized triblock copolymers of polyfp-methyl-styrene) and PDMS and their derivatives obtained by the selective chlorination of the hard segments were investigated by TEM 146). Samples with low PDMS content (12%) showed spherical domains of PDMS in a poly(p-methylstyrene) matrix. Samples with nearly equimolar composition showed a continuous lamellar morphology. In both cases the domain structure was very fine, indicating sharp interfaces. Domain sizes were estimated to be of the order of 50-300 A. 

Two-mirror telescopes are the most common optical design for ground based telescopes. These systems require a parabolic or hyperbolic primary mirror. As mentioned before, more complex optical systems can accommodate a spherical primary with its attendant simplifications, but several additional mirrors are needed to correct the spherical aberration, and the light loss and alignment complexity makes this configuration less commonly used. Here we will assume that a non spherical primary is needed and we will discuss the resulting surface shapes that segments will have. 

With spherical segments, optical figuring and testing is a proven and reliable process, well suited for mass-production. Serial production of diffraction-limited, large optics is already under way for laser fusion projects, with European suppliers increasing their capacity to approximately 1,000 m2 per year Aspherical segments would certainly be feasible as well, but the inherent risk and potentially lower quality need to be properly evaluated. In figuring optical... 

The secret is to build a primary that is aspheric to get the desirable optical properties of these designs while keeping the segments as close as possible to a shape that is spherical or can be polished almost as easily as a sphere such as a toroid, a surface with two constant radii. To study this idea further, consider a mirror that is a parabola of revolution. We use a parabola because the more realistic hyperboloid is only a few percent different from the parabola but the equations are simpler and thus give more insight into the real issues of fabrication. The sagitta, or sag, of a parabola is its depth measured along a diameter with respect to its vertex, or... 

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