Combinatorially fruitful properties of 3⋅2 and 3⋅2 modulo p

Write a≡3⋅2⁻¹ and where p is an odd prime. Let c be a value that is congruent (modp) to either a or b. For any x from Z_p?{0}, evaluate each of x and within the interval (0,p). Then consider the quantity where the differences are evaluated in the interval (0,p−1), and the quantity where the differences are evaluated (modp+1) in the interval (0,p+1). As x varies over Z_p?{0}, the values of each of and give exactly two occurrences of nearly every member of 1,2,…,(p−1)/2. This fact enables a and b to be used in constructing some terraces for Z_p−1 and Z_p+1 from segments of elements that are themselves initially evaluated in Z_p.     

Structure of minimum thickness and terraced free-standing films of block copolymers

The morphologies of thin, substrate-free block copolymer films have been examined by cross-sectional TEM. Two poly(styrene-b-butadiene) diblock copolymers were studied: one that forms PS cylinders and the other that forms PB cylinders in the bulk. Films were annealed while supported by metal TEM grids, embedded, and ultramicrotomed in crosssection. We find that at the metal support the film forms a meniscus-like region, or Plateau border, which exhibits the bulk morphology. Away from the border, the film thickness decreases and regions of terraced in-plane cylinder domains occur until a minimum thickness is reached. The minimum thickness region of the PB majority copolymer in cross-section shows a PS interlayer penetrated by a hexagonal array of circular PB channels that connect upper and lower PB surface layers, and a total thickness of 25–27 nm. The minimum thickness region of the PS majority copolymer in plan view shows no image contrast, but in cross-section reveals a continuous PS interlayer covered by layers of PB, and a total thickness of 20 nm. Comparisons with the chain dimensions suggest a bilayer arrangement for both morphologies with strongly perturbed chain conformations in the surface layers. © 1996 John Wiley & Sons, Inc.     

Terraces on banded spherulites of polyhydroxyalkanoates

On banded spherulites of poly(3‐hydroxybutyrate) and its copolymers, fine circles were observed between either cross‐polarizers or without polarizers. Atomic force microscopy was applied to confirm that these circles are terraces with heights up to several hundred nanometers rather than cracks. Real‐time observation demonstrated that the terrace forms at the front of the growing spherulites just before or exactly when two spherulites impinge on each other. Terraces were observed on the spherulites crystallized from melt confined between glass or polyimide slides rather than poly(ethylene terephthalate) slides. The formation of the terraces may have resulted from instability of the moving boundary of the melt film confined between the spherulite surface and cover slide. Wettability of the substrate played an important role in the formation of the terraces. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2128–2134, 2003