cascading phenomenon can be attributed to theRayleigh instabilityof the charged
liquid droplets, which explains the splitting of the charged droplets due to the
overcrowding of surface charges (Rayleigh 1882 ). Several smaller droplets split
from a big one could effectively reduce the density of charges on the high-energy
surface of the droplets.
Assuming that each collapsed bead containsmchain monomers, with the linear
sizeR~m1/3, and the fraction of charged monomers isp, with one charged
monomer containing one charge, the surface free energygR^2 will be balanced
with the charge repulsive energy (epm)^2 /(eR) in each collapsed bead. By
minimizing their total free energy with respect toR, we obtain
R^3 ðpmÞ^2 (4.64)Fig. 4.11 Illustration of parallel stacking of polyelectrolyte chains (dark grayrods with negative
charges) favoring the sharing of counter-ion clouds (light graypositive charges) and hence
forming liquid-crystal-ordering structure
Fig. 4.12 Illustration of the cascading collapse transition of a charged polymer in a poor solvent,
with a bead-string structure that eventually collapses into a single sphere
4.3 Single-Chain Conformation in Polyelectrolyte Solutions 63