Polymer Physics

and the ionic strength for the salts in solution is

I¼Sz^2 iri (4.61)

in whichirepresents ion species,zirepresents the number density, andrirepresents
the charge valence (Israelachvili 1985 ). The strong screening interactions of the
massive ions enable the polyelectrolyte chain in the salt-added solutions to behave
like a neutral polymer chain in a typical non-polar solvent.
Owning to the electrostatic repulsions along the chain, polyelectrolyte chains
obtain an additional semi-flexibility beyond the neutral polymers, appearing as the
additional persistence lengthlelec that has been calledOdijk-Skolnick-Fixman
persistence length.

l¼l 0 þlelec (4.62)

and

lelec¼lBð

lDH

2 d

Þ^2 (4.63)

wheredis the average distance between the charged groups along the chain (Odijk
1977 ; Skolnick and Fixman 1977 ; Odijk and Houwaart 1978 ). In practice,lelecis
much larger than the Debye-Hu ̈ckel screening length, suggesting that the polyelec-
trolyte chain mostly behaves like a rigid rod. Poly(acrylate acid) chains are flexible
neutral polymers in the weak acidic solutions. However, with the increase of pH
value, for instance, by adding sodium hydroxide, the polymer chains become
sodium Polyacrylate and behave like rigid polyelectrolyte chains. Such a pH-
responsive function can potentially be utilized in mimicking biological switches
or stretching of muscles.
The counter-ion concentrations around the polyelectrolyte chain may fluctuate,
which induces an attractive interaction between the chains sharing the same clouds
of counter-ions (Golestanian et al. 1999 ). Parallel stacking of polyelectrolyte chains
favors such kind of attractions, as demonstrated in Fig.4.11. This effect will result
in the spontaneous liquid crystal ordering in polyelectrolyte solutions (Potemkin
et al. 2002 ; Potemkin and Khokhlov 2004 ). Such a tendency of liquid crystal
ordering actually stabilizes the parallel rolling of DNA long chains, and squeezes
them into the very limited space of cell nucleus.
In a poor solvent, a single polyelectrolyte chain will collapse into a sphere similar
to a charge-neutral chain. However, due to the existence of Coulomb interactions, the
collapse transition appears slightly more complicated. As demonstrated in Fig.4.12,
the single chain containing high density of charges first collapses into a bead-string
structure. With further decrease of the temperature,lBwill gradually rise up, and the
apparent charge density will decay. Correspondingly, the number of beads on the
string will decrease as well, causing the coil to shrink through several cascading steps
till reaching the size of a single condensed sphere (Dobrynin et al. 1996 ). Such a

62 4 Scaling Analysis of Real-Chain Conformations