Linear charge density, polyelectrolyte-counterion

Horvath J, Nagy M (2006) Role of linear charge density and counterion quality in thermo dynamic properties of strong acid type polyelectrolytes divalent transition metal cations. Langmuir 22 10%3 10971... 

Formula (2.3.41) implies that in a system with added electrolyte of a higher valency z, the magnitude of the effective linear charge density is determined by the latter. Upon reduction of the concentration of the added electrolyte (N —> 0) a transition occurs in the way prescribed by (2.3.41) to the effective linear charge as determined by the proper counterions of valency Q in a polyelectrolyte solution free from added low molecular electrolyte. 

Several chapters of this book discuss applications and extensions of the theory of polyelectrolyte solutions. Counterion condensation theory postulates that for a cylindrical macroion, if the linear charge density exceeds a well-defined critical value, a sufficient fraction of the counterions will "condense" into the immediate domain of the macroion so as to reduce the net charge density due to the macroion and Its condensed counterions to the critical value. No condensation is predicted for macroions with less than the critical charge density. 

A dimensionless linear charge density, is normally used as a convenient parameter for counterion condensation and polyelectrolyte theories. It is defined as... 

This asymptotic behavior is a consequence of the different distribution of counterions on the middle and at the extremes of the chain. Therefore, as the size of the chain increases, end-effects become less important. Polyelectrolytes with a greater linear charge density achieve a larger 0p " value at a larger chain size j ateau End effects are more important in this case, due to the larger counterion condensation in these systems. The plateau values are 0 o.78 0.63 0.43... 

We intend to present first a set of experimental data concerning the equilibrium properties (pK, activity of counterions) of polyelectrolytes in salt free solutions in a second part, these results are compared with theoretical values obtained when a rod-like model is adopted in the Lifson-Katchalsky, Manning and Oosawa treatments. The polyelectrolytes tested are respectively polysaccharides (alginic and pectic acids), carboxymethylated derivatives of cellulose (CMC), dextran (CMD), amylose (CMA), and sulphonic derivatives of polyvinyl alcohol. The essential difference between them is local rigidity. It is possible to vary by synthesis the degree of substitution and consequently the linear charge density our data are given for dilute aqueous solutions (2 X 10 N-5 X 10" N) at 25°C. 

From a general point of view, the distribution of counterions is imposed by that of the electrostatic potential. Our purpose is to treat the thermodynamic behaviour of a cylindrical system without excess added salt the polyelectrolyte is always considered as a thin rod characterized by its linear charge density. Two approaches are investigated the first one needs the resolution of a Poisson-Boltzmann equation without... 

In polyelectrolyte solutions, the counterion condensation on linear polyelectrolyte chains is known to occur when the charge density along the chain exceeds the critical value [40]. Our work indicates the existence of a critical value for the separation distance between chains, where the interchain interaction changes drastically, most likely due to the transition in the binding mode of the counterions (see Fig. 13). Many peculiar forms of behavior, which are often interpreted by the cluster formation or the interchain organization of polyelectrolytes, have been reported for high concentrations of aqueous polyelectrolytes... 

Manning proposed a linear counterion condensation theory to account for the low activity of counterions in polyelectrolyte solutions 14). The basic idea of the theo is that there is a critical charge density on a polymer chain beyond which some counterions will condense to the polymer chain to lower the charge density, otherwise the ener of the system would approach infinite. The concept of this theory has been widely accepted. The shortcoming of the linear countmon condensation is that it predicts that counterion condensation is independent of ionic strength in the solution, which is not in agreement with experimental observations. Counterion condensation can be obtained duectly by solving the nonlinear Poisson-Boltzmann equation. 

---