Chain Swelling by Excluded Volume Effect

In addition to effects arising from local stiffness along the chain, monomers of real chains undergo excluded volume interactions among themselves mediated by solvent molecules as pointed out in Section 2.3. Since the chain connectivity has been parametrized in terms of Kuhn steps, we define the second virial coefficient (v) for a pair of Kuhn segments, in an equivalent way to Equation 2.17, as 

Therefore, the repulsive excluded volume interaction favors larger values of R (by lowering Fgve), and the entropic part due to chain connectivity favors smaller values of R (by lowering Fconnectivity)- As a result, an optimum is attained. This is obtained by minimizing F with respect to R. Leaving out the coefficients, the result is 

For strong repulsive excluded volume interactions, corresponding to good polymer solutions (T 6p), the size exponent is 3/5, called the Flory exponent Vp 

The above result for the coil size (Equation 2.73) crosses over smoothly between the Gaussian and Flory limits. 

A similar formula to the above can be derived for the radius of gyration. The qualitative features and the exponents in the asymptotic regimes are the same, the only difference being a slight variation in the numerical coefficients. The factor 4/3 on the right-hand side of Equations 2.73 and 2.75 must be replaced by 134/105 (Yamakawa 1971). 

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