imply that determination of the molar mass of a polyelectrolyte via osmotic
pressure should be done at very high ionic strength and requires
extrapolation to very smalljvalues. (It may be noted that some other
complications arise, but these can be taken into account.)
6.4.2 Chain Overlap
When the concentration of polymer in solution is increased, the coiled
molecules effectively fill the whole volume at a certain critical concentration.
This was briefly discussed in the Question in Section 6.2.1. At still higher
concentration, chain overlap occurs; in other words, the molecules
interpenetrate, becoming mutually entangled. This is illustrated in Figure
6.12 for a neutral polymer.
Polymer scientists distinguish variousregimes, as shown in Figure
6.13, which gives an example of a state diagram; actually, it represents quite
a small part of the diagram, albeit the part that is the most relevant at the
moment. The variables are the (net) volume fraction of polymerjand the
excluded volume parameterb(or the solvent-segment interaction parameter
w). In the domain or regime called dilute, the relations discussed in Section
6.2 hold; going from A via B (‘‘ideal’’) to C, the molecules decrease in extent
of expansion. Both in the ‘‘semidilute’’ and the concentrated regime, there is
TABLE6.3 Osmolality of Polymer Solutionsa
j 0.003 0.01 0.031st virial term
n¼ 100 0.75 2.50 7.5
n¼ 1000 0.075 0.25 0.75
2nd virial term
b¼ 0000
b¼ 0 : 5 0.125 1.39 12.5
2nd virial term
mc¼ 1 126 1400 12,600
mc¼ 10 12.6 140 1260
3rd virial term 0.005 0.02 0.5aGiven are the virial termsðmol?m (^3) Þcalculated according to Eq. (6.11).
The osmotic pressure in Pa thus equalsRT&2500 times the sum of the
virial terms. The second virial term is given for a neutral polymer (main
variableb) and for a polyelectrolyte (main variablemc). Molar volume of
solventVs¼ 18? 10 ^6 , of polymerVp¼ 4? 10 ^5 nm^3 ?mol^1 .n¼number
of monomers per molecule. For the polyelectrolyte the valence is 0.3n, and
mc¼concentration of counterionsðmol?m^3 Þ.