Manning model, counterion condensation

Manning GS. Counterion condensation theory constructed from different models. Physica A 1996 231 236-253. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

Consider a counterion of absolute valence Zq that is held fixed at a distance r from the polyion. The polyion is modeled within the framework of the Manning line model [31] discussed in Section Chapter II.A. The idea here is that the reduction of a polyion charge is dictated by how far the counterion is from that charge. Hence, the renormalized charge due to counterion condensation is written as qnet = (1 — %(r))q, where q is the bare charge [43]. 

The Manning counterion condensation line model provides insights into the mean-square number fluctuations of the condensed counterions, ((A0)2), where 0 is the number of condensed counterions per polyion charge [50], Denote by 0O the value of 0 for which the total polyelectrolyte free energy G per charge is a minimum, that is, (0G/00)e=e vanishes. Expanding G in powers of 0 0O to quadratic order leads to [50]... 

Manning GS. Limiting laws and counterion condensation in polyelectrolyte solutions. V. Further development of the chemical model. Biophys Chem 1978 9 65-70. 

A theoretical approach for the second Wien effect of polyelectrolytes has been initiated in terms of the counterion condensation model by Manning. According to this theory the degree of counterion dissociation from the condensed layer is linearly dependent on the field strength. 

In the Manning model, if > 1, the DNA is thermodynamically unstable and the counterions will condense on it until reaches the value of unity. When = 1, l=b theoretically this situation corresponds to an extended DNA backone-and b corresponds almost to the Landau length. In the temperature range 20-80°C, the Landau length does not vary... 

Manning determined values of fc and Ap for a line charge model with counterion condensation (18). Above fc. 

Manning GS (1996) Counterion condensation theory constructed from different models. Physica A 231(l-3) 236-253. doi 10.1016/0378-4371(95)00452-l... 

Some of the most successful computer simulation studies of nucleic acid systems to date employ models without explicit treatment of counterions, using instead the Manning model for counterion condensation. 4 Manning developed the counterion condensation model from polyelectrolyte theories applied to DNA treated as a charged cylinder. This model, resulting from a series of studies 2,43 is summarized by ... 

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