Average counterion concentration

Noticing that the average counterion concentration is Nc for fully ionized polyelectrolytes, we obtain... 

FIG. 1 Variation of the local concentration, C, of the monovalent counterion of radius 0.3 nm with the radial distance, r, from the axis of the cylindrical polyion of radius a = 0.8 nm. The average counterion concentration is marked by Cav, d is the exclusion distance from the polyion axis to the center of the counterion, and/is the fraction of free counterions. The value of the charge density parameter A = 2.83 is typical for aqueous solutions of poly(styrenesulfonates) at 25°C, and the value of the radius of the cell R = 65.2 nm corresponds to the polyelectrolyte concentration CP = 5 X 10 4 monomol dm3. 

In Fig. 6, the star size R and the excess concentration of counterions at the outer cell boundary, r = D, are presented as a function of the number of arms, p, for different values of a (shown by solid lines). Dotted lines in Fig. 6b, indicate the corresponding expectations for uniform distribution of the cell counterions. An increase in the number of arms in each star implies an increase in the number of charged monomers Q = paN, and in the corresponding number of mobile counterions in the cell. Figures 6 and 7 clearly demonstrate a transition from a barely charged to an osmotic star behavior upon the increase in p. At a relatively small number of arms, the star size and the concentration of counterions at the outer cell boundary grow as a function of p. The latter is approximately proportional to p and is close to the average counterion concentration in the cell. This proves that ions... 

The average counterion concentration is found by converting the numbers no and Ns to concentrations keeping in mind that the volume accessible to the site (1/Cs) is larger than that accessible to the counterion ... 

Following this same procedure, but with variable 8, also shows that 8 = i if the potential is to remain finite at r = 0. We now find y by applying the bovmd-ary condition on the electric field at r = R but first note that electroneutrality requires that the average counterion concentration be related to the surface charge density by... 

Second, we have to be cautious regarding the physical significance of Eqn. 11. The parameter, jS, is an average property, and it may be that with surfactants such as CTAOH, or CTAF, in water, we have a mixture of very small micelles which only weakly bind counterions, and larger, normal , micelles which bind them strongly, and with increasing surfactant or counterion concentration the population shifts from small to normal micelles and increases. When the counterion is less hydrophilic, e.g., Br , the normal micelles would predominate. 

Surface tension isotherms for some sodium salts of fatty acids (Cn02Na) are shown in Fig. 3.41 sodium decanoate (n=10), laurate (n=12) and myristate (n=14). Here the experimental data of [33, 65] are presented, where NaOH [65] or NaOH + NaHCOj [33] were added to the solutions of C 02Na to prevent the solutions from hydrolysis. In both cases the concentration of added sodium ions was 0.1 M. This counterion concentration was introduced in the calculations of the average ionic product as c = f (c x+xy rx ... 

We assume that net surface charge density (5) and similarly die fraction of bound counterions to NaPAA, are is independent of counterion type. Thus the total concentration of Na" and l-ArN2 counterions is 65% of that of the pendent carboxylate groups, ti = 0.65, Over a 5-fold polymer dilution, the average interfacial counterion concentration in the condensed volume, Ccv around NaPAA is about 0.14 M estimated by chemical trapping. 

Fig. 48. Comparison of the aggregation number (Ag i) of a mixture of conjugated bile salts and the range of aggregation numbers of trihydroxy and dihydroxy bile salts. At each different counterion concentration (NaCl) the average size of the mixed micelle lies between the size of the trihydroxy and dihydroxy bile salt micelles.

Consider a positively charged polyelectrolyte chain (see Figure 9). Let us place the center of the coordinate system at the chain s center of mass with the z-axis pointing along the direaion of the chain elongation. Counterions with valence cj = -l are distributed around a polyion with average local concentration qon(f). The distribution of counterions aeates distribution of the electrostatic potential P(r), which satisfies the Poisson equation ... 

Figure 4.2 (a) A snapshot of a flexible polyelectrolyte with its counterions for N = 60 at r = 2.8 and the average segment concentration 5.63 x (b) Profiles of the electric potential, monomer density, and the counterion density against the radial distance from the center of mass of the chain. 

Here, c is the average molar negative ion (counterion) concentration. Hence, the Poisson equation reduces to... 

One attraction of MD simulation is the possibility of computer animation. The mobility of ions inside a charged cylindrical pore can be visualized. Some movie clips of EMD and NEMD are downloadable at http //chem.hku.hk/ kyc/movies/. mpg. Some features that escape statistical averages can be learned in watching the animation. While the coions are present mainly in the center of the pore, occasional collisions with the wall do occur, as observed in the movie. The time scale of a coion staying near the wall is of the order of 1 ps, compared to 10 ps for the counterion. While the averaged equilibrium distributions indicate an infinitesimal concentration of coion at the wall, reaction of coion with the wall can occur within a time scale of 1 ps. From the video, it can also be observed that the radial mobility of the counterion is more significant compared to the coion s and compared to the axial mobility. It is consistent with the statistical results. 

Experiments were performed at 5°C in order to arrest the cis-trans isomerization of the protonated Schiff base. Spectra with one equivalent of acid and different mixing times showed one NOE cross-peak between H15 of the retinal molecule and the proton on the counterion, as shown for a mixing time of 0.4 s in Figure 10. The strong chemical shift dependence of the H15 resonance on the concentration of the acid dictated the use of less than one equivalent of the protonating formic acid, and therefore an incomplete protonation (>80%) of the retinal, in order to avoid an overlap between the formate and the H15 peaks in the spectrum. This should not affect the observed result since an average chemical shift, between those of HI 5 of the retinal in its nonprotonated and protonated... 

The effect of monomer concentration on number average molecular weight indicates that while transfer to monomer occurs within an ion-counterion-monomer complex with an activation energy equal to that of propagation, termination occurs in an uncomplexed ion-counterion pair and decreases relative to propagation with decreasing temperature. 

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