Unoccupied volume fraction

Figure 13.13. Glass transition temperature of ethyl cel- Figure 13.14. Unoccupied volume fraction in ediyl lulose vs. weight fraction of tributyl citrate. [Data cellulose vs. weight fraction of hibutyl citote. [Data...

Figure 13.13 shows that the glass transition temperature of ethyl cellulose decreases along with increase in fraction of tributyl citrate. Figure 13.14 shows that the unoccupied volume fraction increases with the addition of the plasticizer as it does with increase in temperature. Figure 13.15 shows that permeability of gas increases when more plasticizer is added. Figure 13.16 shows that the concentration of the plasticizer determines the rate of the active substance released from pharmacentical tablets designed for controlled release. This set of data shows the major reasons for use of plasticizers in the pharmaceutical industry. 

Figure 16.1. Unoccupied volume fraction in ethyl cellulose films plasticized with variable concentrations of tributyl citrate. [Data fi-om Beck M I Tomka I, J. Polym. Sci. Polym. Phys. Ed., 35, No.4, March 1997, p.639-53.]...

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. 

In the derivation of the mean-field partition function, it is necessary to know the probability for inserting a chain molecule in a given conformation into the system. The classical way to compute this quantity is by approximating it by a product of the local volume fractions of an unoccupied site (averaged over lattice layers). It was realised that, besides the density information, information on the bond distributions is also available. The bond distribution gives information on the average local order. Using this information, it becomes possible to more accurately access the vacancy probability. 

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 ... 

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. 

First we will work out Pj. Once the first sub-particle s position is determined each of other sub-particles must be the closest neighbor to the preceding sub-particle shown in Figure 2.4. In addition, the first unit must be immediately next to the last unit of the preceding sub-particle. The probability of such an arrangement is the volume fraction of unoccupied cells in the system, so that... 

Fig. 49a. A representative configuration of block copolymers on the lattice (For clarity a square lattice is shown, while all work refers to a simple cubic lattice). Three symmetric diblock copolymers are shown, each of chain length N = 10. The two monomeric species are labeled A-type (full dots) and B-type (open dots). The vacancies are not shown explicitly, but are assumed to reside on each lattice site left unoccupied by either of the two species of monomer. A volume fraction of < >v = 0.2 is used, since experience with blends [107] has shown that such a system behaves like a very dense melt. The energy contributions eAA, eBB and eAB are shown, b Examples of typical slithering-snake [298,299] motion monomer situated at point labelled by 5 is removed, and one of sites 1,2,3 is randomly chosen for occupation. Note that unlike Refs. [298,299] also the junction point needs to be displaced accordingly, as shown in the figure. For the reverse process, monomer at 3 is removed and the sites 4,5,6 are considered for attachment (of course, a move to site 6 is rejected due to excluded volume constraints), c Interchange of A-Block and B-Block of a diblock copolymer chain. From Fried and Binder [325],...

Some values ofr) and o are shown in Table 13.1.2 including the two extreme cases. Actually, water and n-hexadecane have the lowest and highest packing density, respectively, of the common solvents. As is seen, there is an appreciable free volume, which may be expressed by the volume fraction 13 -11 , where 13 0 is the maximum value oft) calculated for the face-centered cubic packing of HS molecules where all molecules are in contact with each other is tIq = nV2 / 6 = Q74. Thus, 1 -13 0 corresponds to the minimum of unoccupied volume. Since 13 typically is around 0.5, about a quarter of the total liquid volume is empty enabling solvent molecules to change their coordinates and hence local density fluctuations to occur. 

Figure 16.1 shows that unoccupied volume increases with increase in plasticizer fraction. Figure 16.2 shows that the diffusion coefficient of oxygen also increases with plasticizer fraction. 

The free volume can be defined simply as the volume unoccupied by the mucromolecules (the occupied volume contains both the van der Waals volume of the atoms and the excluded volume, see also chapter V). In the classy state (T < Tj) the free volume fraction v, is virtually constant. However, above the glass transition temperature... 

The problem of determining the theoretical behavior of polymers in the glassy rates is treated by Curro et al. (34,35). The time dependence of the volume in the glassy state is accounted for by allowing the fraction of unoccupied volume sites to depend on time. This permits the application of the Doolittle equation to predict the shift in viscoelastic relaxation times. 

The Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

The most successful statistical theory of liquids is that derived by Simha and Somcynsky. The model considers liquids to be mixtures of voids dispersed in solid matter, i.e., a lattice of unoccupied and occupied sites. The occupied volume fraction, y (or its counterpart the free volume fraction f = 1 - y), is the principal variable y = P, T). From die configurational partition function the configurational contribution to the Helmholtz molar free energy of liquid i was expressed as [3] ... 

Williams, Landel, and Ferry (WFF) observed that if Tr is set to Tg, the variation of log flr with T — Tr is similar for a wide variety of polymers [10]. They rationalized this in terms of the molecular response, starting with Doolittle s equation [Eq. (41)] for the viscosity, where A and B are constants. f is the fractional free volume, equivalent to the unoccupied volume divided by the total volume of the polymer (the occupied volume includes that necessary to accommodate thermal vibrations). 

In Fig. 15.9, the dependence of on mean distance r between chaotically distributed in amorphous PC radicals-probes is adduced. For PC at T = 77K the values of djd = 0.38 0.40 were obtained. One can make an assumption about volume fractions relation for the ordered domains (nanoclusters) and loosely packed matrix of amorphous PC. The indicated value djd means, that in PC at probes statistical distribution 0.40 of its volume is accessible for radicals and approximately 0.60 of volume remains unoccupied by spin probes, that is, the nanoclusters relative fraction according to the EPR method makes up approximately 0.60 0.62. 

This has been done for polyethylene, polyisobutylene and polypropylene models, for instance, by prescribing spheres of a chosen diameter around each interaction site, or by using Voronoi polyhedra to define the unoccupied space. The free volume fraction,can be defined by... 

A maximum fractional unoccupied volume f ) was defined by Litt et al. [92] through the densities of the amorphous (pg) and the crystalline state (Pc) as ... 

According to free-volume interpretations, the rate of molecular motions is governed entirely by the available unoccupied space ( free volume ). Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, / [23,24]... 

The pellets leave a fraction e unoccupied as they pack into the reactor so the fraction 1 — is occupied by the catalyst. The pellet is usually porous, and there is fluid (void space) both between catalyst pellets and within pellets. We measure the rate per unit area of pellet of assumed geometrical volume of pellet so we count only the void fraction external to the pellet. 

Is this the correct initial pressure to use Or should we account for the internal void of the adsorbent as well when we compute the initial pressure. To do so would lead to one more term in volume, namely, that of the void fraction within the solid. This is not the void between the solid particles, but that which is within the solid particles. If the mass of the particles is ms and their density is ps, then the volume of the particles is Vp = and if the fraction that is unoccupied by solid is then this extra volume is Vp = C The corrected initial pressure would be ... 

The S-S theory describes the structure of a liquid by a lattice model with cells of the same size and a coordination number of z = 12. The disordered structure of the liquid is modeled by allowing an occupied lattice-site fraction y=y(V,73 of less then 1. The configurational or Helmholtz free energy, F, is expressed in terms of the volume V, temperature 7] and occupied lattice-site fraction y = y(K7), F=F(V,T,y). The value of y is obtained through the pressure equation P = —(9F/9V)r and the minimization condition (dFldy)v,T = 0. The hole fraction is given by the fraction of unoccupied lattice sites (holes or vacancies), which is denoted by h, h(P,T) = —y(P,T). This theory provides an excellent tool for analyzing the volumetric behavior of linear macromolecules but was also applied successfully to nonlinear polymers, copolymers, and blends. Several universal relationships where found which allow an approximate estimation of the fraction of the hole (or excess) free volume h and the total or van der Waals free volume/ [Simha and Carri, 1994 Dlubek and Pionteck, 2008d]. For more details, see Chapters 4, 6, and 14. 

In Equations 2.41 and 2.42, the left-hand sides denote the rates of adsorption, which are proportional to the pressure and the unoccupied surface. The right-hand sides denote the rates of desorption, which are proportional to the occupied surface. The rate constants are a for adsorption and [b exp(- QIRT) and b exp(-QJRT)] for desorption in the two cases (n = 1 and n > 1). 0 is the fractional area covered by the -stack layer. The volume adsorbed per unit area is proportional... 

The basic assumptions of the kinetic-molecular theory give us insight into why real gases deviate from ideal behavior. The molecules of an ideal gas are assumed to occupy no space and have no attractions for one another. Real molecules, however, do have finite volumes, and they do attract one another. As shown in Figure 10.25 , the free, unoccupied space in which molecules can move is somewhat less than the container volume. At relatively low pressures the volume of the gas molecules is negligible, compared with the container volume. Thus, the free volume available to the molecules is essentially the entire volume of the container. As the pressure increases, however, the free space in which the molecules can move becomes a smaller fraction of the container volume. Under these conditions, therefore, gas volumes tend to be slightly greater than those predicted by the ideal-gas equation. 

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