Introduction to diffusion, swelling, and drying

The free-volume theory of diffusion was developed by Vrentas and Duda. This theory is based on the assumption that movement of a small molecule (e.g., solvent) is accompanied by a movement in the solid matrix to fill the free volume (hole) left by a displaced solvent molecule. Several important conditions must be described to model the process. These include the time scales of solvent movement and the movement of solid matrix (e.g. polymer segments, called jumping units), the size of holes which may fit both solvent molecules and jumping units, and the energy required for the diffusion to occur. 

The timescale of the diffusion process is determined by the use of the diffusion Deborah, number De, given by the following equation  

If the diffusion Deborah number is small (small moleeular relaxation time or large diffusion time) moleeular relaxation is mueh faster than diffusive transport (in fact, it is almost instantaneous). In this case the diffusion process is similar to simple liquids. For example, diluted solutions and polymer solutions above glass transition temperature fall in fliis category. 

If the Deborah number is large (large molecular relaxation time or small diffusion time), the diffusion process is described by Fickian kinetics and is denoted by an elastic diffusion process. The polymeric structure in this process is essentially unaffected and coefficients of mutual and self-diffusion become identical. Elastic diffusion is observed at low solvent concentrations below the glass transition temperature.  

The relationships below give the energy required for the diffusion process and compare the sizes of holes required for the solvent and polymer jumping unit to move within the system. The free-volume coefficient of self-diffusion is given by the equation  

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Preliminary drying of PEMs increases adsorption of the dyes. The immersion of dried PEM into dye solution causes its swelling due to both solvent and dye molecules introduction into the film material, while adsorption in wet PEM occurs by slow diffusion of a dye in already swollen material. 

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