Microscopic theory of semidilute solutions

To explain the behavior of polymer solutions in the overlapped regime, it is necessary to adopt a picture of the structure of these systems. The chain molecule is represented as a string of m statistical subunits with mean-squared end-to-end vectors R12 The average concentration inside a statistical subunit can be expressed as  

One arbitrary definition of the semidilute regime is the concentration range c c c.  

As the chains start to overlap, tiie individual subrmits still tend to avoid one another. A useful quantity introduced by deGennes is the intramolecular pair-correlation function for subrmits inside a single chain. Consider a particular subunit at the origin surrounded by a sphere of radius r. The concentration of polymer inside the sphere can be calculated if we can determine the number of subunits that, on average, will be foimd within the sphere. For a good solvent, the mean-squared end-to-end distance of a chain, here taken to be scales as vf fu) This means that the average concentration within a sphere of radius r scales as  

There is no longer a tendency for subunits in different regions to be repelled, and it is convenient to view the overall chain as a sequence of subregions of mean-squared end-to-end length that can be described by random-coil statistics. The number of such subregions should scale as m/n. The mean-squared end-to-end length of the chain should now scale as  

As the screening length decreases in a good solvent, the chain dimensions should also decrease until the screening length is comparable to the statistical subunit length. 

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