Thermal expansivity of free volume

This choice relies on the assumption that a constant Tg+T corresponds to a constant free volume state. Such an approximation presumes that the thermal expansion of free volume, Uf, as well as the fractional free volume, fg, at Tg, are independent of the hlend composition. In the case of binary blends with short chains, may depend on the composition of the blend, so that a normalisation of experimental times with this parameter would be misleading. 

Kilburn et al. [2002] carried out a PALS study of free volume in semicrystalline poly(ethylene-co-l-octene) (PO) copolymers as well as high-density polyethylene (HDPF). The degree of crystallinity was characterized by DSC and WAXD analyses. A method was proposed to estimate the fractions of the RAF and MAF phases based on the observation that the mean thermal expansivity of free-volume holes, ea = d vh)/dt, varies as a function of Xc, which implies that the individual mean expansivities of holes in RAF and MAF phases are different. Thus, the thermal expansivity of the mean hole volume (i.e., averaged over the entire amorphous phase) may be expressed above To by... 

In addition, free volume increases with increasing T because of thermal expansion. The free volume can therefore play a role in thermal expansion processes both at low and at high temperatures. The detailed discussion of free volume is outside the scope of this book. See Robertson s excellent review article [10] for a thorough discussion of this concept. 

Ferry and his co-workers have given further consideration to the exact form of the WLF equation. It can be shown that a better fit to data for different polymers can be obtained by changing the constants Cf and C and that the actual values obtained for Cf and C yield values for fg and a/, which are plausible on physical grounds. The reader is referred to Ferry s book [15] for detailed discussion of these points. We will, however, note here that the fractional free volume at the glass transition temperature fg is 0.025 0.003 for most amorphous polymers. The thermal coefficient of expansion of free volume Of is a more variable quantity, but has the physically reasonable universal average value of 4.8 X 10-4 K-i. 

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

We propose to rationalize the observation by a phenomenon known as residual thermal stresses. Residual thermal stresses arise from the fact that carbon-fiber and epoxy have different thermal expansion coefficients and a quenching of the composite would conceivably produce residual stresses. Apparently, the quenching process may produce enough residual stresses to lower the toughness of the composite. In the absence of such residual stresses the free volume concept alone would predict a quenched glass to have larger amount of free volume and hence constitute a less brittle substance. 

Alternative ways of determining the free volume fraction without using I3 have also been proposed by Dlubek et al [28], as well as, Brandzuch et al [29], Dlubek et al used the coefficient of thermal expansion of the amorphous regions and hole volume determined from positron data to determine the number density of the free volume holes. Brandzuch et. al. used the coefficient of thermal expansion just above and just below the Tg to estimate the fractional free volumes. This model is based on the assumption that the expansion of the holes of the free volume, as seen by positrons, reflects the expansion of the total volume of the material. 

Fractional Free Volume and Coefficient of Thermal Expansion. The shift constants c and C2 from the WLF equation are not only fitting parameters that describe the frequency-temperature relation of a given polymer, but they are also related to chemical structure. Ferry has shown (6) that these constants can be related to the fractional free volume and coefficient of thermal expansion of the free volume, which have physical meaning in terms of the polymer structure. One can define the free volume at the glass transition divided by the total volume as fg and the coefficient of thermal expansion of... 

The thermal expansivity of crystalline solid is evaluated via calculation of its Helmholtz free energy, A(T, V), which is a function of the temperature, T, and the volume, V. The free energy is expressed by... 

To test the Rouse prediction that viscosity is proportional to chain length, viscosity data at constant friction coefficient must be used instead of viscosity data at constant temperature. If the coefficient of thermal expansion of the free volume af m Eq. (8.131) were independent of chain... 

If the difference between the coefficients of thermal expansion of the solvent above and below Tgs is unknown, Equation 3.7 can be used to provide a first estimate by substituting the quantities referring to the solvent instead of those referring to the polymer into this equation. This procedure amounts to making the assumption that the free volume arguments underlying Equation 3.7 are just as valid for simple molecular liquids as they are for amorphous polymers. If this assumption is made, Equation 6.12 is simplified into Equation 6.13 which should only be used if the necessary thermal expansion data are unavailable for the solvent. 

In contrast to the compressibility, the thermal expansion of proteins has received much less attention. Frauenfelder et al. [65] have estimated the thermal expansion of myoglobin from the refined X-ray structure at 80 and 255-300K. They conclude that the expansion comes mainly from the subatomic free volumes between the atoms. The expansion obtained from the temperature dependence of the vibrational frequency shifts of the hydrogen bonds support... 

Polymer-solvent mixtures can be separated and the polymer recovered from solution at the lower critical solution temperature (LCST). This is the temperature at which the miscible polymer-solvent mixture separates into a polymer-rich phase and a solvent-rich phase. LCST phenomena are related to the chemical nature of the mixture components, the molecular weight of the mixture components, especially the polymer, and the critical temperature and critical pressure of the solvent (Allen and Baker, 1965). As the single-phase polymer solution is isobarically heated to conditions near the critical point of the solvent, the polymer and solvent thermally expand at different rates. This means their free volumes change at different rates (Patterson, 1969). The thermal expansion of the solvent is much greater than that of the polymer. Near its critical point, the solvent has expanded so much that it is no longer able to solubilize the polymer. Hence, the polymer falls out of solution. If the molecular weight of the polymer is on the order of 10 a polymer-solvent LCST can occur within about 20-30°C of the solvent s critical temperature. If the molecular weight of the polymer is closer to 10, the LCST phase... 

The concept of free volume has been of more limited use in the prediction of solubility coefficients although, Peterlin (H) has suggested that the solubility coefficient is directly proportional to the free volume available in the polymer matrix. In many respects, the free volume expressions closely resemble the relationships developed in the activated state approach. In fact for the case of diffusivity, the two models can be shown to be mathematically equivalent by incorporating thermal expansion models such as the one proposed by Fox and Flory (12). The usefulness of the free volume model however, lies in the accessibility of the fractional free volume, through the use of group contribution methods developed by Bondi (12.) and Sugden (li), for correlation of barrier properties of polymers of different structure as demonstrated by Lee (15.). ... 

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