Relation of Molecular Mobility to Free Volume

The appearance of the glass transition results from the reduction of molecular mobility as the temperature falls, slowing the collapse of free volume. We now introduce the concept that the mobility at any temperature depends primarily on the free volume remaining, so that the rates of both bulk and shear deformations can be advantageously expressed in terms of Vf rather than T as the independent variable. This principle was applied long ago to the shear viscosity of simple liquids by Batchinski, and more recently by Doolittle with an empirical equation which was found to represent with high accuracy viscosities of ordinary liquids of low 

The physical basis of equation 29 can be understood from the theory of Cohen and Turnbull,which treats the self-diffusion of spherical molecules with the assumption that motion of a molecule can occur only when a void exceeding some critical volume v , is available for it to move into. If no energy is required for free volume redistribution at constant volume, a calculation of fluctuations gives a result which is equivalent to expressing the translational friction coefficient f as follows  

Here should be proportional to T -y is a numerical factor between 0.5 and 1, and Vfm is the average free volume per molecule. From relations such as equations 44,51, or 55 of Chapter 10, the viscosity should have essentially the same temperature dependence as f. Hence equations 29 and 30 are equivalent, provided (a) the slight temperature dependence of and the factor 1 / T can be neglected (b) v jvf s (u - Vf)lvf, i.e., the minimum void size is close to the occupied volume per molecule (c) B and y are negligibly different from unity, or equal (d) in the case of polymers, v and v/m refer to volumes per some kind of segmental unit instead of volumes per molecule. (Other derivations of equations similar in form to equation 29 have been given by Bueche. ) 

Equations 29 and 30 imply that free volume is the sole parameter in determining the rate of molecular rearrangements and transport phenomena such as diffusion and viscosity which depend on them. In older theories of liquids, - the temperature dependence of viscosity is determined by an energy barrier for hole formation. This leads to a viscosity proportional to 6 p AH RT), where A//, is the activation energy for flow, independent of temperature—an Arrhenius form. It will be shown in Section 6 that the latter type of temperature dependence is applicable at temperatures very far above Tg.- whereas equation 29 is applicable for 100 or so above Tg, and hybrid expressions may also be useful over a more extended range. 

To express equation 29 in terms of the shift factor a-p, we employ equation 13, together with the definition/ = oy/p, and obtain 

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