Measurement Techniques and Their Mathematics

This section is concerned with measurement techniques of the diffusivity and solubility from which the permeability can easily be calculated. In the following analysis we restrict ourselves to the measurement of constant values of D. Concentration- and position-dependent diffusivities are analyzed in Crank and Park (1968) and Crank (1975). Generally, the techniques are for permeability, steady-state and time lag techniques and for diffusivity, sorption and desorption kinetics and concentration-distance curves. For self-diffusivity in polymer melts the techniques are (Tirrell, 1984) nuclear magnetic resonance, neutron scattering, radioactive tracer, and infrared spectroscopy. 

In the steady-state permeation method, the surfaces of a polymer film of thickness i = 2b are kept at constant gas pressures Pao an PAr- At steady state Eq. 4.75 applies, and the permeability P is found from this equation. If the solubility is known, then the diffusivity can be found from Eq. 4.71. For hollow cylinders and spheres, expressions similar to Eq. 4.74 hold. On the other hand, in the time lag method we deal with the unsteady-state permeation of a diffusant through a slab of thickness 2b. The surfaces of the slab are kept at concentrations Cao and at zero. The accumulated amount of gas which has passed through the slab in time t. 

At long times (i.e., when steady state is achieved) the exponential term becomes negligible, so that 

The intercept of the above equation with the time axis gives the so-called time lag as 

In the sorption kinetics techniques, the mass uptake of a slab of thickness 2b at time t. 

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