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The charge of a polyelectrolyte (PE) controls myriads of phenomena inbiology, biotechnology, and materials science, but still remains elusive froman understanding. Considering the adsorption of counterions on an isolated PEchain, an analytical expression for the effective charge - valid for all chainflexibility, for variable salt, in good solvents at all conditions, in poorsolvents in the expanded (coil) state away from the critical point, and forgels, brushes and other PE systems in their expanded state - is derived. Phaseboundaries and the critical point for the I-st order collapse transition,induced cooperatively by counterion adsorption and chain conformations, arecalculated self-consistently. The size of the PE chain is found to be asingle-valued function of charge.

Effective charge, collapse and the critical point of a polyelectrolyte chain