Chain Collapse and Counterion Condensation in Dilute Polyelectrolyte Solutions

A quantitative theory for polyelectrolytes in salt-free dilute solutions is developed. Depending on the electrostatic interaction strength, polyelectrolytes in solutions can undergo strong stretching (with polyelectrolyte dimension $R_{\mathrm{g}}\sim l_B^{1/3}N$, where $l_B$ is the Bjerrum length and $N$ is the number of the chain segments) or strong compression (with $R_{\mathrm{g}}\sim l_B^{-1\mathrm{}/2}{N}^{1/3}$). A strong polymer collapse occurs as a first-order phase transition due to accompanying counterion condensation.