Free volume equilibrium fractional

In the first row the relative dielectric constant for the compound is given. In the second row the valency of the unit is given. The other rows give the values for the various FH parameters. Remaining parameters the characteristic size of a lattice site 0.3 nm the equilibrium constant for water association K — 100 the energy difference for a local gauche conformation with respect to a local trans energy it/ 8 — 0.8 A T the volume fraction in the bulk (pressure control) of free volume was fixed to (pbv = 0.042575... 

The conclusion that the free-volume fraction at Tg is not a universal parameter for linear polymers of differing molecular structure can be qualitatively confirmed by the following arguments71. Assume that at temperatures far below Tg polymeric chains are in a state of minimum energy of intramolecular interaction, Le. the fraction of higher-energy ( flexed ) bonds is zeroS4. On the other hand, let the equilibrium fraction of flexed bonds at T> Tg obey the Boltzmann statistics and be a function of Boltzmann s factor e/kT. Thus, the fraction of flexed bonds at Tg can be estimated from the familiar expression ... 

As the glass or liquid ages and its volume decreases, the fictive temperature also decreases, finally reaching the actual temperature T if and when the glass or liquid reaches thermodynamic equilibrium. From Figs. 4-10 and 4-17, it can readily be deduced that Tf can be related to the fractional free volume by ... 

There is no fundamental qualitative difference in mechanisms of low molecular weight (MW) penetrant diffusion in polymers above and below glass transition temperature, Tg, of the polymers [5,6]. The difference lies only in the fact that the movement of structural units of the macromolecule that are responsible for the transfer of penetrant molecules takes place at different supermolecular levels of the polymer matrix. At T > Tg the process of diffusion takes place in a medium with equilibrium or near-equUibrium packing of chains, and the fractional free volume, P(, in the polymer is equal to the fractional free volume in the polymer determined by thermal mobUity of strucmral units of macromolecules V((T), i e., V(= vut). At r< Tg the process of diffusion comes about under nonequihbrium packing conditions, although there exists a quasi-equilibrium structural organization of the matrix, where Vf> It is assumed that in this case Vf= where is the fractional free volume... 

The free-volume model was originally derived to explain the temperature dependence of the viscosity. We have shown that it has a much broader application and can explain many of the outstanding experimental observations. This includes the existence of an entropy catastrophe at 7 and the approximate equality of Tj, and 7], first observed by Angell and coworkers.The relation between ln and 7, measured by Moynihan et al., also follows naturally and quantitatively from the notion that the liquidlike cell fraction p is the important variable that ceases to reach equilibrium when the relaxation rates become longer than the time scale for the measurement. 

Obtaining F in this manner, the first matter is computation of the free-volume fraction h= 1 - y. At equilibrium this is done by the minimization of F at the V and T specified ... 

FIGURE 4.14 Distribution of free volume in PVAc immediately after the temperature was stepped to 30°C after equilibrium had been established at 40°C (wq) and after equilibrium is attained at 30°C (ILco)- Shown is the relative fraction of states having the free volume indicated by the abscissa (x). 

Robertson et al. [1984] developed a stochastic model for predicting the kinetics of physical aging of polymer glasses. The equilibrium volume at a given temperature, the hole fraction, and the fluctuations in free volume were derived from the S-S cell-hole theory. The rate of volume changes was assumed to be related to the local free volume content thus, it varied from one region to the next according to a probability function. The model predictions compared favorably with the results from Kovacs laboratory. Its evolution and recent advances are discussed by Simha and Robertson in Chapter 4. 

Total deviation of the fractional free volume from equilibrium... 

A semi-grand canonical treatment for the phase behaviour of colloidal spheres plus non-adsorbing polymers was proposed by Lekkerkerker [141], who developed free volume theory (also called osmotic equilibrium theory ), see Chap. 3. The main difference with TPT [115] is that free volume theory (FVT) accounts for polymer partitioning between the phases and corrects for multiple overlap of depletion layers, hence avoids the assumption of pair-wise additivity which becomes inaccurate for relatively thick depletion layers. These effects are incorporated through scaled particle theory (see for instance [136] and references therein). The resulting free volume theory (FVT) phase diagrams calculated by Lekkerkerker et al. [142] revealed that for <0.3 coexisting fluid-solid phases are predicted, whereas at low colloid volume fractions a gas-hquid coexistence is found for q > 0.3, as was predicted by TPT. 

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