Swelling of Crossslinked Polymers

A three-dimensional network polymer, such as vulcanized rubber, does not dissolve in any solvent. It may nevertheless absorb a large quantity of a suitable liquid with which it is placed in contact and undergo swelling. The swollen gel is essentially a solution of solvent in polymer, although unlike an ordinary polymer solution it is an elastic rather than a viscous one. 

A suitable expression for AGmix in the present case may be obtained from Eq. 

A suitable expression for AGei derived (Flory and Rehner, 1943, 1950) by analogy with the deformation of rubber under the condition of isotropy is 

Problem 3.20 The structure of a three-dimensional random network may be described quantitatively by two quantities the density of crosslinking designated by the fraction e of the total structural units engaged in crosslinkages and the fraction 6f of the total units which occurs as terminal units or free chain ends (i.e., which are connected to the structure by only one bond). Alternative quantities, such as the number (mole) N of primary molecules and the number (mole) v of crosslinked units, in addition to M and Me, defined above, are also used to characterize a random network stmeture. Relate N and v to these other quantities. 

The chemical potential of the solvent in the swollen gel is obtained, by adding Eqs. (3.120) and (3.121) and partial differentiation with respect to ni in order to apply Eq. (3.3). Thus, 

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