Coarse-Grained Sheet

The morphology and shape of a sheet play crudal roles in its spedfic functions. A platelet or tethered membrane [33, 40-44] exhibits unique conformational characteristics as it wrinkles and crumples under various physical constraints, such as quality of solvent and temperature, while entropy dissipates from its open boundaries. As pointed out above, one can tether partides - the smallest units in a coarsegrained description - to form both sheets and other hierarchical units such as chains. 

We know that the stochastic motion of a particle exhibits a range of dynamics (anomalous diffusion, diffusion, and drift) from short to long asymptotic time regime. Let us consider the motion of a random walk of a particle on a cubic lattice in which a particle hops to one of its nearest-neighbor sites (six on a cubic lattice) randomly at each time step. The probability that the random walker reaches a distance R from the origin (the starting point) in f hops (or time steps) is P(t,R) = 

A exp(—R /Cjt), where A and Q are constants. The root mean square displacement (R) can be evaluated by integrating it with this distribution and provides R= with a diffusion constant C2. The motion can be explicitly monitored in a computer simulation and the power-law relation between the rms displacement R) and the time step (t) can be verified. 

A stochastic move is defined as follows select a particle at a site i and select one of its 2 6 neighboring sites, sayj. If sitej is empty and the constraint on the changes in the bond lengths (due to the proposed move) to all cormected particles is satisfied, then the particle is moved from site i to sitej. On the other hand, if either of the criteria is not met, the particle remains at its original site i. Attempts to move each particle once defines the unit Monte Carlo step (MCS) time. A simulation then consists of a large number of such (MCS) steps, typically millions. 

The excluded volume, self-avoiding sheet (SAS) deforms and wrinkles propagate as its particles execute their stochastic motion. The conformation and dynamics of the sheet can be studied by keeping track of the rms displacement of the center of mass of the sheet and that of its center (interior) particle, radius of gyration, as a function of molecular weight and rigidity of the covalent bonds. Effects of the quality of the solvent and entanglement barriers (e.g., polymer chains matrix) can be incorporated (see below). A few observations from our computer simulations are briefly discussed in the next section. 

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In order to dry the crystals, the Buchner funnel is inverted over two or three thicknesses of drying paper (i.e., coarse-grained, smooth surfaced Alter paper) resting upon a pad of newspaper, and the crystalline cake is removed with the aid of a clean spatula several sheets of drying paper are placed on top and the crystals are pressed flrmly. If the sheets become too soiled by the mother liquor absorbed, the crystals should be transferred to fresh paper. The disadvantage of this method of rapid drying is that the recrystallised product is liable to become contaminated with the Alter paper flbre. 

In the case of sheet and strip for drawing or stamping, coarse grain may give a rough surface. On the other hand, metal with too fine a grain size may lack plasticity and crack in the dies therefore, a compromise must he reached. 

The nitrator contents are drawn off to the filter tank (1) (Fig. 110). The tank is lead-lined and equipped with a ventilating pipe for conducting away nitrogen oxide. Filtration is most effective with a 20-cm layer of coarse-grained dinitronaphthalene as a filter material, placed between perforated plates of sheet aluminium. 

Fig. 18. 23 This small rock sample was collected on January 18, 1982 by Ian WhiUans and John Schutt on the Middle-Western ice field on the East Antarctic ice sheet west of the Allan HiUs. It was subsequently identified as a lunar rock by scientists at the Johnson Space Center in Houston, Texas, and was assigned the identifying number ALHA 81005. The rock contains fragments of coarse-grained plagioclase feldspar in a fine-grained black matrix and has been described as an anorthosite breccia from the highlands of the Moon. (Courtesy ofNASA/LPI))...

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