Counterion condensation on ionic oligomers

The Ramanathan-Woodbury formulas representing the charge density critical for the onset of counterion condensation on finite-length polymers are derived by three alternate methods, an extension of Debye-Huckel theory, a theory of end effects, and by density functional theory. For charged oligomers with length of the same order as the Debye length, the threshold for condensation is the same as for polymers of length much greater than the Debye length. However, the threshold depends both on length and salt concentration if the oligomer is shorter than the Debye length, in such a way as to recede to infinity as the ratio of oligomer length to Debye length tends to zero (i.e., condensation vanishes in this limit). The extended Debye-Huckel theory additionally provides a new result for the partition function of the condensed layer, showing that the free energy of the condensed counterions is different on an oligomer and a polymer, even when the fractional extent of condensation is the same. The end effect theory discloses a hitherto unnoticed connection between the number of counterions condensed at the ends of a long polymer and the number condensed on a short oligomer.