Free volume temperature dependence

If an amorphoiis polymer is cooled it will usually attempt to crystallize, but because of the high internal viscosity of the medium it is often precluded from packing into its lowest energy conformation. At 0 K, the lack of thermal excitation prevents the occurrence of most photochemical reactions. As the temperature is increased, the specific volume of the polymer will also increase as a result of forming "free volume", that is, space vdiich is not occupied by hard-shell dimensions of the atoms comprising the polymeric structure. The amount of free volume will depend to a certain extent on the previous thermal history. As free volume increases along with thermal excitation, various kinds of molecular motions will be observed in the polymer vdiich can be detected by I ysical measurements. 

Fig. 22. Schematic representation of the volume temperature dependence, lowing the expansion factor for the glass, a, the liquid, oci, and the occupied volume, o%, and the fractional free volume / at the glass temperature T ...

At large concentrations of polymer the zero-shear viscosity of the mixture can be written as the product of two parameters a monomer friction coefficient and a structure factor F (1). The friction factor is controlled by local features such as the free volume and depends on the temperature. The structure factor is controlled by the large-scale structure and the configuration of the polymer chain. This factor depends on the dimensions and the molecular weight of the polymer chains and the polymer concentration. If the friction factor is properly evaluated for concentration dependence, the structure factor F depends for concentrated solutions on the polymer concentration to the power 3.4 (1-3). Therefore the rheological properties of a polymer-monomer mixture are mainly determined by the amount and the molecular weight of the polymer in the mixture. These different concepts can be combined to a semiempirical model for the viscosity (4) ... 

The fluorescence properties of these probes permits us to study the rotational relaxation in various polymers and even during polymerization reactions and thereby obtain information on the microscopic rigidity of the media. In the following discussion a description of the photophysical properties of the dyes 1-3 will be given, with particular emphasis on the excited-state conformational relaxation in various media. This will be followed by a discussion related to the application of these probes to study polymerization reactions, the effect of polymer molecular structure on free-volume, the dependence of polymer chain relaxation on molecular weight, and the effect of temperature on polymer conformation and free-volume. 

In discussing Fig. 4.1 we noted that the apparent location of Tg is dependent on the time allowed for the specific volume measurements. Volume contractions occur for a long time below Tg The lower the temperature, the longer it takes to reach an equilibrium volume. It is the equilibrium volume which should be used in the representation summarized by Fig. 4.15. In actual practice, what is often done is to allow a convenient and standardized time between changing the temperature and reading the volume. Instead of directly tackling the rate of collapse of free volume, we shall approach this subject empirically, using a property which we have previously described in terms of free volume, namely, viscosity. 

The iatroduction of a plasticizer, which is a molecule of lower molecular weight than the resia, has the abiUty to impart a greater free volume per volume of material because there is an iucrease iu the proportion of end groups and the plasticizer has a glass-transition temperature, T, lower than that of the resia itself A detailed mathematical treatment (2) of this phenomenon can be carried out to explain the success of some plasticizers and the failure of others. Clearly, the use of a given plasticizer iu a certain appHcation is a compromise between the above ideas and physical properties such as volatiUty, compatibihty, high and low temperature performance, viscosity, etc. This choice is appHcation dependent, ie, there is no ideal plasticizer for every appHcation. 

The rate of solvent diffusion through the film depends not only on the temperature and the T of the film but also on the solvent stmcture and solvent-polymer iuteractions. The solvent molecules move through free-volume holes iu the films and the rate of movement is more rapid for small molecules than for large ones. Additionally, linear molecules may diffuse more rapidly because their cross-sectional area is smaller than that of branched-chain isomers. Eor example, although isobutyl acetate (IBAc) [105-46-4] has a higher relative evaporation rate than -butyl acetate... 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

A more rigorous free-volume treatment is due to Cohen and Grest (CG) [34,35], according to which the material is comprised of liquid and solid-like cells. The former have free volume, but mobility requires continuity of the local empty space. The temperature dependence of the relaxation times according to the CG model is... 

Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. I. Reexamination of the Free-Volume Theory, Journal of Polymer Science Polymer Physics Edition 15, 403, 1977. Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecnlar Weight, Journal of Polymer Science Polymer Physics Edition 15, 417, 1977. 

Figure 2 The temperature dependence of the self-diffusion coefficient of 2,3-dimethyl-butane predicted by Cohen and Turnbull s free volume model. (From Ref. 25.)...

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