Line charge model

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

The simplicity and implication of the above results can be illustrated by considering a line charge model of dsDNA. Let the charge separation length I be 0.17 nm, so that F = 4.2 at 20 Let Zp = 1 and Zc = 1. Therefore, for conditions where counterions would condense (FzpZc > 1), the fraction of condensed counterions, I — a, follows from Equation 3.94 as... 

However, it must be emphasized that these results are only mathematical artifacts, as the exact solutions for the potential and counterion distribution, for the line charge model with a finite radius, do not show any such phase-transition-like discontinuities (Alfrey et al. 1951). The three major objections to the line-charge model of flexible polyelectrolyte are zero thickness, infinite length, and no-chain flexibility. 

The extrapolation procedures, based on Equation (13), are applicable to all thermodynamic properties which could be derived from the cell model. It is necessary to emphasize that the limiting expressions obtained in this way are equal to those following from the line charge model. 

It is evident that the idea of the cell model with cylindrical symmetry, introduced years ago by Prof. Katchalsky to the field of polyelectrolytes, has been very successful. It has promoted the experimental research and initiated the development of other models having a cylindrical symmetry. Today, we would like to have a deeper understanding of the reasons for the wide applicability of this model. It seems that at this stage no satisfactory explanation can be expected. We believe, however, that the line charge model will be the first and the cell model the second approximation of a future more elaborate theory of polyelectrolyte solutions. 

Figure 4.8 Potential-dependent reaction energies for water dissociation to form OH, O, and H over Pt(l 11). (a) Energy curves based on the full charge model the nonlinearity of these plots expresses the capacitance of the interface, (b) Differences of the curves indicate the reaction energies. The nonlinear terms cancel almost completely. The dashed lines indicate predictions made from the linear model, whereas the solid lines are predictions made from a fuU solvation/ charge-based model [Rossmeisl et al., 2006].

Fig. 1.7 Evolution of COD and ICE inset) in function of specific charge for different organic compounds (cross) acetic add, (open square) isopropanol, (open circle) phenol, (open triangle) 4-chlorophenol, (open diamond) 2-naphtol i = 238 A m-2 T = 25°C Electrolyte 1M H2SO4. The solid line represents model prediction...

Fig. 49. Comparison of SCF and point-charge model values of W(r) in the molecular plane for H2O. Solid line refers to SCF values, dashed lines to the model. From Ref. 57)...

Figure 23 Ab initio effective charge product (Eqs. [59] and [60]) in the D2/, water dimer (top). Filled squares, Qo,o, diamonds, Qo,Hj- The solid line is a least-squares fit to the O1O2 points. The coefficients of the curve are shown in the figure. The theoretical values for the intercept and slope, based on atomic multipoles of the water monomer, are 0.6194 and 1.341, demonstrating that the electrostatic properties of water dimers are described adequately by an atomic multipole model. Interestingly, the usual point-charge model would have predicted a line with zero slope (Reprinted with permission from ref. 120, copyright 1989 American Institute of Physics.)...

This figure compares with an electrical barrier of 0.28 eV estimated from the point charge model. These values would coincide for a = 0.34. Hence the theory based on the Verwey-Overbeek potential is consistent with experiment. From (4) and (5) the experimental plot of In v versus AF should yield a straight line with the so-called Liley slope. Inserting the numerical values, we find that this slope, with a = 0.34, corresponds to a tenfold increase in frequency of MEPP for each 15.0 mV depolarization. The most recent experimental value on the rat muscle preparation was 12.5 mV, and Liley reported about 16 mV. We can thus conclude that a discussion of the approach of a synaptic vesicle to a presynaptic membrane, which is based on the Verwey-Overbeek theory of the interaction of two double layers, gives reasonable quantitative agreement with the available data on the dependence of V of MEPP on depolarization of the presynaptic membrane. 

The charge model has been used in formulating a strategy for infill drilling to develop additional reserves within the Judy Field. Petroleum-water contacts and fluid types are in line with pre-drilling predictions. 

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