Osmotic pressure for polyelectrolytes-, singular behaviour

Though the study of linear polyelectrolytes is beyond the scope of this book, their behaviour is solution differs so much from the behaviour of neutral polymers, that it is worthwhile to mention a few striking properties of these solutions. 

A polyelectrolyte is a polymer which, dissolved in water, ionizes and gives one poly-ion and small-size counter-ions. Thus, sodium polystyrene sulphonate is a typical poly-electrolyte. 

The number of free charges carried by the chain is called the ionization degree Z and the number of Na counter-ions simultaneously produced also equals Z (where Z N). 

It can be shown that an isolated polymer, in solution in pure water is always completely ionized (Z = V) then the polyion has the shape of a nearly rigid rod. This entails that the solutions that are currently used are nearly always semi-dilute, i.e. in these solutions the chains overlap one another. 

The membranes used to measure the osmotic pressure are impermeable to polyions but permeable to counter-ions. In spite of this, the osmotic pressure of polyelectrolytes in pure water and for rather small concentrations (but in the semi-dilute regime), is huge. Let us assume that the polyelectrolyte has been put in cell I (see Fig. 5.1). It partially ionizes but both cells must remain practically neutral. The counter-ions which, theoretically, can cross the membrane, are retained in cell I and a contact-potential difference appears at the boundary between the cells. Actually, it looks as if the counter-ions contributed like polyions to the osmotic pressure. Let C be the polyion concentration. We may write approximately 

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