Manning s theory

Plots of the bursting pressures of the Ni—Cr—Mo cylinders (EN 25) vs k derived from equations 16 and 17 show that neither equation is in such good agreement with the experimental results as is the curve derived from Manning s theory. Similar conclusions have been reached for cylinders made of other materials which have been tested (16). Manning s analytical procedure may be programmed for computation and, although torsion tests are not as commonly specified as tension tests, they are not difficult or expensive to carry out (20). 

According to Manning s theory [43-45], a polyion is characterized by a dimensionless charge density parameter B, which is defined by... 

Manning s theory does not take the local effective dielectric constant into consideration, but simply uses the a value of bulk water for the calculation of E,. However, since counterion condensation is supposed to take place on the surface of polyions. Manning s 2, should be modified to E, by replacing a with aeff. The modified parameters E, is compared with E, in Table 1, which leads to the conclusion that the linear charge density parameter calculated with the bulk dielectric constant considerably underestimates the correct one corresponding to the interfacial dielectric constant. 

According to Manning s theory, Na+ ions condense onto the polyanion until the fraction (1 — ) of the line charge is neutralized when , > 1 i.e. the effective... 

The theoretical relationships for fa and Dj/D, ° from Manning s theory [102-108] as well as from other theories, for example, of Iwasa [114], Gueron [115], andYoshida [116,117] have been summarized in Table 9. 

In [17], the electric polarizability y = 2 X 10 30 Fm2 of an adsorbed sodium polystyrenesulfonate (NaPSS) is found almost equal to y = 1.5 X 10 30 Fm2 of NaPSS in solution with no added salt [49], Strangely, this value is in good concordance with the one estimated from Manning s theory, even though the experimental values are relevant to fast polarization of the counterionic atmosphere. 

We carried out potentiometric investigations with a Cl sensitive electrode [40] to determine the degree of conversion, i.e., the release of the low molecular counterions in the reactions between sodium poly(styrene sulfonate) (NaPSS) and poly(diallyldimethylammonium chloride) (PDADMAC) and its copolymers with acrylamide of various compositions (for synthesis and characterization of the samples see [41]). The basic idea of these studies on PEC formation is the change of the Cl ion activity coefficient due to the release of the counterions. According to Manning s theory [31], the activity coefficient of the counterions of a polyelectrolyte is given by... 

The Combined Stern-Volmer and Perrin Model A model has been proposed by Morishima et al. [97] which takes account of Manning s theory [98] of polyelectrolytes and introduces a modification into the Stem-Volmer equation to describe sphere-of-action (Perrin) quenching this has been termed combined Stern-Volmer and Perrin Analysis and has been adopted [95,96] in an effort to describe quenching of fluorescence from labeled PMAA by T1+ ions, for example. 

The sodium and calcium activity coefficients were determined by specific electrodes, in dilute salt-free solutions (c < 10 equiv/I) of sodium and calcium pectinates. Results were compared to those calculated from Manning s theory (21). This model is proposed for infinitely dilute solutions of rodlike polyelectrolytes and the activity coefficients are directly imposed by the charge parameter ... 

We have also shown previously (12l by measuring D + when CaCl2 is added to ChSO.-Na that the small fraction (15 percent) of the sodium ions initially condensed are expelled from the condensation layer by the newly condensing Ca " " ions, which can be expected from Manning s theory. 

Figure 2. Conq)arison of experimental salt absorption with theoretical predictions based on ideal Donnan equUibrium (Cal-I), nonlinear counterion condensation theory (Cal-N), linear counterion condensation theory (Manning s theory) (Cal-L)...

Transport parameters (/) of divalent counterions for pectins with different DE. Comparison of experimental values with the theoretical one calculated with Manning s theory [92]. 

According to Manning s theory for polyelectrolytes [72], a charged macromolecule in solution is submitted to counterion condensation when the so-called charge parameter, 4, is higher than unity, enabling this parameter to decrease. 

The properties of polyelectrolyte solutions depend strongly on the interactions between the polymers and the surrounding counterions. Manning s theory of counterion condensation predicts that a certain quantity of counterions condenses onto a polymer, whose charge density exceeds a critical value [44]. This leads to an effective decrease in the polymer charge. The macroscopic properties of the polyelectrolyte are not determined by its bare charge but by an effective charge. In particular, the flexibility and hydrophobicity of the polyelectrolyte chain, the... 

According to Manning s theory [40], the rodlike polyelectrolyte can capture N oppositely charged counterions ... 

Number of moles of dissociated repeating units Number of moles of counterions originating from P (in Manning s theory)... 

Molar volume in Manning s theory Molar volume of pure solvent s Configurational parameter Configurational parameter... 

For highly diluted solutions, the results of Manning s theory agree with the results of Lifson and Katchalsky [cf. (46) and (47)]. For example, Manning [106] gives for the osmotic coefficient of an aqueous solution of a single polyion where the counterions have the (absolute) charge munber zq ... 

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