Area-minimizing surfaces

Thomas EL, Anderson DM, Henkee CS et al. (1988) Periodic area-minimizing surfaces in block copolymers. Nature 334 598-601... 

On the basis that interfacial tension constitutes the dominating factor for structure formation in microphase-separated block copolymers, Thomas et al. [87] have proposed that the complex nanostructures formed in block copolymers correspond to area-minimizing surfaces. From the extensive SCFT calculations, Matsen and Bates [94, 95] find that an equally important, but thus far disregarded, factor in block copolymer nanostructure stability is packing frustration [96]. For the minority blocks of an ordered copolymer to fill space uniformly, the interface self-adjusts so that no blocks are excessively stretched. This entropic consideration causes the interface to deviate from CMC (with ct 0), in which case Oh provides a measure of packing frustration and nanostructural stability. Although predicted (H) and Oh are only available for diblock copolymers [95] (which differ from the present triblock copolymer in molecular architecture), it is worthwhile to compare... 

Particle shape (for a given volume, the best shape for minimizing surface area and improving filterability is a sphere), and... 

The net attraction of surface molecules to the interior of the liquid indicates that molecules are most stable when attractive forces are maximized by as many neighbor molecules as possible. Consequently, a liquid is most stable when the fewest molecules are at its surface. This occurs when the liquid has minimal surface area. Spheres have less surface area per unit volume than any other shape, so small drops of a liquid tend to be spheres. Large drops are distorted from ideal spheres by the force of gravity. 

In direct insertion techniques, reproducibility is the main obstacle in developing a reliable analytical technique. One of the many variables to take into account is sample shape. A compact sample with minimal surface area is ideal [64]. Direct mass-spectrometric characterisation in the direct insertion probe is not very quantitative, and, even under optimised conditions, mass discrimination in the analysis of polydisperse polymers and specific oligomer discrimination may occur. For nonvolatile additives that do not evaporate up to 350 °C, direct quantitative analysis by thermal desorption is not possible (e.g. Hostanox 03, MW 794). Good quantitation is also prevented by contamination of the ion source by pyrolysis products of the polymeric matrix. For polymer-based calibration standards, the homogeneity of the samples is of great importance. Hyphenated techniques such as LC-ESI-ToFMS and LC-MALDI-ToFMS have been developed for polymer analyses in which the reliable quantitative features of LC are combined with the identification power and structure analysis of MS. 

The surface area of a spill should be minimized for hazardous materials that have a significant vapor pressure at ambient conditions, such as acrylonitrile or chlorine. This will make it easier and more practical to collect vapor from a spill or to suppress vapor release with foam or by other means. This may require a deeper nondrained dike area than normal or some other design that will minimize surface area, in order to contain the required volume. It is usually not desirable to cover a diked area to restrict loss of vapor if the spill consists of a flammable or combustible material. 

FFs that are parameterized for high-pressure conditions can still lead to behavior that differs from that observed in experiments. For instance, it is common practice to treat the interatomic interactions with Lennard-Jones (LJ) potentials. Although this method is convenient from a computational standpoint, it is known that LJ potentials do not reproduce experimentally observed behavior such as necking, where a material attempts to minimize surface area and will break under large tensile stresses. Many other examples exist where particular types of FFs cannot reproduce properties of materials, and once again, we emphasize that one should ensure that the FF used in the simulation is sufficiently accurate. 

The discontinuous phase generally takes the rough shape of a sphere to minimize surface area exposure to the other phase. The size of the spheres influences the overall properties and varies with concentration. In general, because of the affinity of like polymer chains, spheres tend to grow. Larger sphere sizes are promoted because they give less relative contact area with the other phase. 

Water is not strongly attracted to a wax surface, which is nonpolar. So as to minimize surface area, the water tends to form a sphere. Sitting on a solid surface, however, the spherical drop of water is squashed down into a bead by the force of gravity. 

A vessel whose volume is 235 0 ml, and whose weight evacuated is 13 5217 g + a tare vessel, is filled with an unknown gas at a pressure of 725 torr and a temperature of 19°C It is then closed, wiped with a damp cloth, and hung in the balance case to come to equilibrium with the tare vessel The tare vessel has about the same surface area and is needed to minimize surface moisture effects This second weighing is 13 6109 g + the tare vessel What is the mole weight of the gas7... 

After a liquid spill it may be possible to reduce the surface area by covering the spill or designing drainage systems to minimize surface area. These techniques work by reducing the evaporative surface area. 

The first source of confusion was the fact that minimal surfaces represent local minima in surface area under Plateau (or fixed boundary ) boundary conditions. The importance of this property with respect to cubic phases must be considered to be limited, however, because the surface area of the interfacial dividing surface — drawn between the hydrophilic and the hydrophobic regions of the microstucture - is given simply by the product of the number of surfactant molecules, times the average area per siufactant which is strongly fixed by the steric, van der Waals, and electrostatic interactions between surfactant molecules. Therefore this interfacial area does not in general seek a minimum but rather an optimum value, which does not tend to zero because of the electrostatic repulsion between surfactant head groups. Furthermore,... 

---