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 ${R}_{\mathrm{g}}\ensuremath{\sim}{l}_{B}^{1/3}N$, where ${l}_{B}$ is the Bjerrum length and $N$ is the number of the chain segments) or strong compression (with ${R}_{\mathrm{g}}\ensuremath{\sim}{l}_{B}^{\ensuremath{-}1/2}{N}^{1/3}$). A strong polymer collapse occurs as a first-order phase transition due to accompanying counterion condensation.