Unoccupied volume

Polyisobutylene and similar copolymers appear to "pack" well (density of 0.917 g/cm ) (86) and have fractional free volumes of 0.026 (vs 0.071 for polydimethylsiloxane). The efficient packing in PIB is attributed to the unoccupied volume in the system being largely at the intermolecular interfaces, and thus a polymer chain surface phenomenon. The thicker cross section of PIB chains results in less surface area per carbon atom. 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

Another very important design consideration is that between the point at which the sample is introduced and the point at which it is detected, the dead volume in the equipment must be kept to a minimum. Dead volume means any empty space or unoccupied volume. The presence of too much dead volume can lead to disastrous losses in efficiency (Fig. 2.3c and 5.3b show examples of this). Clearly there will be some dead volume in the column itself, which will be the space that is not occupied by the particles of stationary phase. 

The difference between interstitial species and interpolated species is one of scale. Interstitial sites are thought of as normally unoccupied positions in relatively closely packed crystal structures. One tends to speak of interpolation when foreign atoms enter larger normally unoccupied volumes, cages, or tunnels in a structure, which otherwise remains topotactically unchanged. 

The difference between V- and D-tessilations is as follows each of V-polyhedra includes one network point (or particle) and a void that is closer to this point than to others, each of D-polyhedra includes one cavity and parts of the particles that are the closest to the center of the cavity and all windows that are on the borders with other neighboring cavities. It is convenient to term the latter as PBU/C, where C means cavity. The local coordination number of cavities Zc is equal to the number of the faces of PBU/C (D-polyhedra or D-polygons), and their local porosity e (or eA in 2D space) is equal to the unoccupied volume. Typical D-polyhedra are shown in Figure 9.30 and Figure 9.31. 

These authors were the first FGSE workers to make extensive use of the concept of free volume 42,44) and its effect on transport in polymer systems. That theory asserts that amorphous materials (liquids, polymers) above their glass transition temperature T contain unoccupied volume randomly distributed and in parcels of sufficient size to permit jumps of small molecules — and of polymer jumping segments — to take place. Since liquids have a fractional free volume fdil typically greater than that, f, of polymers, the diffusion rate both of diluent molecules and (uncrosslinked and unentangled) polymer molecules should increase with increasing diluent volume fraction vdi,. The Fujita-Doolittle expression 43) describes this effect quantitatively for the diluent diffusion ... 

Fig. 12 The 2Fob-Fcak omit map used in the automated query. The 2Fobs-Fca c omit map calculated with all reflections to 2.88 A using CNS, map was phased from five shaken high temperature annealed protein models also run at 2.88 A. This omit map exhibits a similar spatial distribution and total volume by comparison with the map in Fig. 11a, but it possesses a single contiguous volume suitable for automated fitting of conformational databases as described in the text. X-ligand was used to locate maximum unoccupied volume of density (highlighted in gold) and to flexibly fit epothilone rotamers from the conformational database. Only two orientations of ligand, as depicted above, were found to satisfy the density shape...

The central point of the present survey is an attempt to show a complete analogy between the free volume of suspensions and that of molecular systems. It is characteristic that the limiting volume fraction of spherical filler particles leaves in the system another 25-40% of unoccupied volume. Precisely the same unoccupied volume exists in molecular systems if we liken them to a volume filled with spheres whose radii are calculated taking into account the Lennard-Jones potential. 

What we have just defined, in a very simplistic maimer, is something called free volume. This is not the same as the empty or unoccupied volume found in the close-packed state (Figure 10-55). Consider the stacking of solid spheres. If we pack these in a regular or ordered fashion there will always be some gaps between the balls. If we allow these spheres to pack randomly, like peas that you throw into a colander to drain, there will be even more of this unoccupied volume, but the spheres will still be close packed, jammed up against their neighbors. However, molecules are not static, like macroscopic balls. Above the absolute zero of temperature they have motion, but in the close-packed state this will just involve vibrations around a mean position. 

So, let us imagine that at very low temperatures in a material that for whatever reason has not or cannot crystallize, the molecules are more or less randomly close packed. The total volume of the system is then that of the hard core of the molecules plus the unoccupied volume between them. As the tem-... 

FIGURE 10-55 Schematic diagram showing the unoccupied volume and oscillations around a mean position in ordered and random close packing of spheres. 

ShaxbyS assumed that, for an ideal liquid, the ratio of the volume occupied by the molecules to the total volume is equal to the ratio of the unoccupied volume to the total volume in the saturated vapour, and deduced a complicated characteristic equation on this basis. 

Figure 3 shows how a sequence of semi-empirical quantum mechanical (QM), force field (FF) and molecular dynamics (MD) calculations can provide a detailed local perspective of the transport phenomena. The information that can be obtained includes the dynamics of the polymer in the absence of the penetrant, the local unoccupied volume... 

The drastic differences of Dqj and from the values observed in PS, and the observation that the density was the only parameter entering the model of P D with a very different value for PVDC than for any of the non-barrier polymers studied, [13,14] indicate that the packing efficiency is the most important descriptive physical parameter both at the "global" and at the "intermediate" scales. The investigation of (i) the local distribution of unoccupied volume, and (ii) the MD trajectory of the penetrant molecule in the polymer matrix, will therefore be very useful. 

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