Fundamental Theory of Swelling

From the viewpoint of statistical mechanics, a similar solution to the one used for a nonideal gas applies. In other words, if the intermolecular potential of a gas is replaced with the elFective potential among solute molecules, which is the interaction transmitted through solvent molecules, the partition ftinction will be in the same form. In the case of a polymer solution, each segment can be regarded as a gaseous molecule. Using this similarity, the theoretical approach developed for a nonideal gas can be applied to polymer solutions. It is easier to understand the excluded volume and 9 temperature fh)m this point of view [1, 2]. 

Any molecule possesses a region where other surrounding molecules cannot invade. This region is called the excluded volume. The fundamental reason for the swelling phenomenon of gels is this excluded volume. If the two-body interaction potential is expressed as U r), the excluded volume is given by  

Using the Letmard-Jones type potential where the expulsion term is replaced by a rigid sphere  

if temperature Tequals u /k, F x is zero. This appears as though the forces cancel each other out in the gas as a whole despite the fact that each molecule impinges on the others with the force dependent on the distances between molecules. It is a strange situation. Equation (1) for the excluded volume is the same equation that gives the second virial coefficient in the virial series of nonideal gases. Therefore, the zero excluded volume is equal to the disappearance of the second virial 

In a polymer solution, the temperature at which = 0 is called the 9 temperature, or more generally the 9 point. At this point, polymer chains behave as though they were ideal Gauss chains. At 0 the polymer chains spread wider than the ideal chains and at 0 they shrink. In a gel where chains are crosslinked, the change of the spreading of an individual chain is observed as macroscopic volumetric changes of the networks. 

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