Geometrical free volume

In the hterature penetrant diffusion has been widely discussed in terms of free volume theories of polymers, and it is not surprising to see this idea taken up in simulation studies. The problem is that although geometric free volume is a physical concept that is easy to comprehend it suffers from lack of precise definition. In the simulation studies to date it has been identified with the unoccupied volume in the sample calculated as an average over many static configurations. Note that this definition is different from the accessible volume to penetrant molecules discussed earlier in this section. 

After construction of the amorphous cell using the method of Theodorou-Suter ° and geometric free volume analysis of the cell several (four to six) penetrant molecules to improve sampling are inserted at the free volume positions. The cell is further relaxed by 100 ps of a NPT-MD (constant particle number, temperature and pressure) simulation at 1 bar and room temperature before starting a longer (nanoseconds) NVT dynamics. The recorded trajectories of each penetrant gas molecule are analyzed and the diffusion coefficient is determined by means of relation (eqn (1.5)). In Figures 1.2 and 1.3 the packed cell model of polydimethylsiloxane (PDMS) and the trajectory of N2 molecules in the PDMS matrix are reported. The MD simulations show two types of motions of the N2 molecules jumps between cavities and local motion inside cavities. 

Figure 4.15 Geometrical representation of the temperature variation of the actual volume (solid line) and the occupied volume (broken line). The shaded difference indicates the free volume which decreases to a critical value at T .

As discussed, the intuitive notion that there should be a connection between the statistics of the free volumes of a fluid and its measurable macroscopic properties has a long history in studies of the liquid state. In fact, it turns out that this connection is precise in the case of the thermodynamics of the single-component hard-sphere fluid. Specifically, Hoover, Ashurst, and Grover77 and Speedy82 have provided independent derivations that predict the relationship between the hard-sphere compressibility factor Z = P/pksT and the geometric properties of its free volumes, as follows ... 

For ultrathin films, it should also be mentioned that Tg is strongly subjected to interface effects, thus it may be higher or lower than in the bulk material [3], This finding can be attributed to the interplay between surface and geometric confinement effects. Due to attractive interactions at interfaces, the molecular dynamics may be slowed down, resulting in an increase of Tg, whereas the confinement to a small layer may lead to an increase in the free volume, resulting in a decrease of Tg [4],... 

Studies by Nishiyama and Fujihara [149] utilizing azobenzene derivative (27) as isomerizable chromophores have demonstrated the importance of reaction cavity free volume in L-B films. The L-B films of amphiphilic derivative 4-octyl-4 -(3-carboxytrimethyleneoxy)-azobenzene (27) upon irradiation was found to be stable, no geometric isomerization of the azobennzene moiety occurred. This compound forms L-B films with water soluble polyallylamine 28 at an air-water interface. Reversible cis-trans photoisomerization occurs in the film containing 28. The reversible photoisomerization reaction in polyion complexed films is thought to occur because of the increased area per molecule provided in the film. The cross sections of molecule 27 in the pure film and in film containing 28 were estimated to be 0.28 and 0.39 nm2. Such an increased area per molecule... 

Figure 27. Inclusion of trans-stilbene in X type and in ZSM-5 zeolites. Required free volume for geometric isomerization is present in supercages of X zeolite and such is absent in ZSM channels. Extensive free volume in ZSM-5 channel is present along the molecular axis, but that is of no use for the reaction to occur.

Analysis of the dependence of viscosity on the concentration of disperse systems and on the free volume of condensed liquid systems shows that there is a considerable similarity between the concepts based on the description of the properties of these systems. This is evidently explained by the similarity of geometric models describing the behavior of these systems based on the description of the... 

The free volume has been introduced intuitively, relating it to the gaps that allow conformational changes in the solid. It would therefore be possible to obtain / from the difference between the geometric volume of the segments and the total volume. Nevertheless, this type of calculation is not useful, as the free volume for molecular movement does not exactly coincide with the empty space in the solid. The concept of free volume is related to the occurrence of macromolecular motion rather than to the existence of gaps. For this reason, the free volume fraction is an empirical parameter whose value is determined on the basis of experimental results. 

The free volume of a liquid, on the basis of this model is thus roughly one tenth of the geometric volume. Alternatively we may write Vf—v-b in (12.47) whence... 

The free volume of a molecule in this picture is to be taken as the geometric volume available to a given molecule contained in its cage This volume is readily calculated in terms of the average distance separating two molecules (d) and their diameter (D). We find that... 

Within the geometric constraints for self-wiping, the OD/ID ratio can be specified to impart a specific average shear rate, define a free volume, or determine allowable shaft diameter (power transmission). While a 2-lobe machine could be designed to have a low OD/ID ratio, as in the 3-lobe unit, this would not be very sensible. The purpose of creating a 2-lobe unit is to have a machine that would be less likely to be volume limited, but still have the power transmission... 

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