Manning condensation

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

The influence of salt on the distribution functions has been discussed within the PB theory in detail in Ref. 4. The general finding was that a low salt content leaves the picture of Manning condensation qualitatively unchanged, while at increasing salt concentration a crossover between Manning condensation and simple salt screening occurs. 

For three systems with numbers of salt molecules N E. 0, 104, 3070 a simulation has been performed and compared to the PB prediction in Figure 10. As in the salt-free case the computer simulations show a more pronounced condensation effect towards the rod. Nevertheless, the shape of the distribution functions remains qualitatively the same. Note in particular that the appearance and disappearance of two points of inflection at N = 104 and N = 3070 respectively, which leads to extremely small curvatures in the PB distribution functions, also leads to very straight regions in the measured distribution functions. The crossover from Manning condensation to screen-... 

Manning Condensation A peculiar phenomenon occurs for highly charged PEs and is known as the Manning... 

In solution, we have considered the scaling behavior of a single PE (Sect. 2.7.3.1). The importance of the electrostatic persistence length was stressed. The Manning condensation of counterions leads to a reduction of the effective linear charge density (Sect. 2.7.3.1.1). Excluded volume effects are typically less important than for neutral polymers (Sect. 2.7.3.1.2). Dilute PE solutions are typically dominated by the behavior of the counterions. So is the large osmotic pressure of dilute PE solutions due to the entropic contribution of the counterions (Sect. 2.7.3.2). Semidilute PE solutions can be described by the RPA, which in particular yields the characteristic peak of the structure factor. 

The characteristic branching parameter (grafting density), n/m = specifies the onset of counterion localization inside the molecular brush. Note that in the osmotic regime, the spacers get fully extended, /t m. It is therefore not surprising, that the counterion localization in a cylindrical molecular brush coincides (in scaling terms) with the Manning condensation threshold [25] for a charged cylinder, qh = 1. 

Mean field theories that implement a spherically symmetric distributions of counterions assume that the (effective) charge density on the star branches is below the Manning condensation threshold. In this limit, one can neglect the angular correlations between the positions of the arms of the star and those of its counterions. 

This length scale characterizes the distance between two charges of the same species allowed by the thermal fluctuations. In other words, when the charge density p on the polymer chain becomes too high, for instance, p > e//, the counter-ions will accumulate around the chain to maintain the effective charge density at constant e//. Such a phenomenon is called Manning condensation of counter-ions (Manning 1969). 

Another result concerns the charge stoichiometry within one complex. For excess DNA, Z /Z = 0.8 whereas for excess polycation Z /Z = 3. It should be noted that these numbers do not represent the effective charge but the chemical charge of the complexes, which may include stericaUy shielded charges in the interior of the cylindrical brushes and ignores possible reduction of charges by counterions (Manning condensation, ion pair formation). 

In the case of strong polyelectrolytes, the number of ionized units corresponds to the number of dissociable ionic units (see Manning condensation) and is independent of the pH. Eor weak polyelectrolytes, the number of ionized units at a given pH is dependent on the p/sTa- Erom acid/base titration, their p/sTa as well as buffering capacity (illustrated by plotting the pH of a solution containing a polymer as a function of the volume of acid added) can be determined. The following equation reported by Patchomik et al. can be used to determine the number of ionized units, i.e., the protonation state of a polycation, at a specific pH [82] ... 

Figure 13 shows the dependence of the reduced effective linear charge density (1 -/3]yo of the rodlike polyelectrolyte on its bare value /o with two qualitatively different regimes. The reduced effective linear charge density (1 -/3]yo increases linearly for small values of the Manning condensation parameter yo... 

Figure 13 Dependence of the reduced effective linear charge density of the rodlike polyelectrolyte on the Oosawa-Manning condensation parameter 70 at different polymer volume fractions . Reproduced with permission from Dobrynin, A. V. Rubinstein, M. Prog. Polym. Sci. 2005, 30,1049-1118. ° Copyright 2005, Elsevier.

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