Fig. 1.

Collected data on diffusion coefficient for the fast (ordinary) mode, $D_f,$ and the slow (extraordinary) mode, $D_s.$ The fast and slow modes emerge for polymer concentrations above a threshold value. For the fully developed two-mode regime, $D_f$ is independent of polymer concentration c and degree of polymerization N over several orders of magnitude and is only about a factor of 4 smaller than the diffusion coefficient of a metallic ion such as Na⁺ in water. $D_s$ is smaller than $D_f$ by three orders of magnitude and it depends on c and N, suggesting formation of aggregates by similarly charged polymers. Upon addition of electrolytes, the slow mode disappears and $D_f$ progressively becomes smaller, approaching eventually the diffusion coefficient $D_{SE}$ expected from the Stokes–Einstein law. Blue triangle, green diamond, red square, black circle, and green triangle data are from Förster et al. (11) for quarternized poly(2-vinylpyridine) with molecular weight $M=1.09\times {10}^5\hspace{0.17em}\text{g/mol},\hspace{0.5em}2\times {10}^5\hspace{0.17em}\text{g/mol},\hspace{0.5em}7.8\times {10}^5\hspace{0.17em}\text{g/mol},\hspace{0.5em}5.8\times {10}^5\hspace{0.17em}\text{g/mol,}$ and $2.26\times {10}^6$ g/mol and degree of quarternization $q=0.65,\hspace{0.5em}0.98,\hspace{0.5em}0.75,\hspace{0.5em}0.4,$ and 1.0, respectively. Purple plus, aqua cross, purple star, and gold circle data are from Sedlak and Amis (15) for sodium poly(styrene sulfonate) with $M=5\times {10}^3\hspace{0.17em}\text{g/mol},\hspace{0.5em}3.82\times {10}^4\hspace{0.17em}\text{g/mol},\hspace{0.5em}1.0\times {10}^5\hspace{0.17em}\text{g/mol,}$ and $1.2\times {10}^6$ g/mol, respectively. The solid lines represent predictions from the present theory.