Donnan potential linearized

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

When ZN/zn [Pg.87]

It is to be noted that when 1, the potential deep inside the surface layer tends to the linearized Donnan potential (Eq. (4.20)), that is. 

Comparison is made with the results for the two conventional models for hard plates given by Honig and Mul [11]. We see that the values of the interaction energy calculated on the basis of the Donnan potential regulation model lie between those calculated from the conventional interaction models (i.e., the constant surface potential model and the constant surface charge density model) and are close to the results obtained the linear superposition approximation. 

As shown in Fig. 28.2, the potential gradient in the membrane core is slightly concave (see curve 3, in particular). However, deviation from linearity is seen to be small in Fig. 28.2. This means that the usual assumption of constant field within the membrane is not a bad approximation. If we employ this approximation, Eq. (28.9) can easily be integrated in the following two limiting cases. When > Hki, l/jCn, in which case the potential far inside the membrane surface layer is in practice the Donnan potential, the constant field assumption yields [3]... 

In the preceding discussion of the relationship of the Donnan potential and the Donnan phase volume terms to the structural parameter of a linear polyion, a/b (A ), it has been shown that the elearostatic effect on the binding equilibria of ionic polysaccharides can be predicted quantitatively by use of the universal curves shown in Figs. 13 and 14. Also, by separating the overall binding equilibria into two processes, i.e., (1) the concentration of counterion in the polyion domain, territorial binding. 

Recently, Mohanty et al. have provided a resolution to this issue [90a]. In this approach, the polyion is regarded as a cylinder of radius R of contour length L immersed in a univalent salt solution. The ions are expected to be bound to the polyion due to the high value of the linear charge density , which for B-DNA is larger than 4. Consequently, the bound ions with the bare polyion are regarded as dressed. The free ions and the dressed ions can be viewed as an ideal mixture that interact via a Donnan potential [90a,91],... 

Approximation (1) is a bad one despite the fact that it leads to simple mathematical solution The concentration profiles are not linear. The partitioning of species between the gel and the sample (2) is also related to the existence of the Donnan potential (7) but it is a problem even for electrically neutral species (e.g. oxygen). If the solution is stirred the effect of the depletion layer at the gel/membrane interface is negligible (3). However, it could be a problem in stationary solutions. Approximations (4) through (6) would be the most... 

The osmotic pressure measurements of micellar solutions has shown that the Donnan potential at the micelle-water interface varies linearly with the added electrolyte concentration, as expected from theory [2,3,13,23]. 

In this section we present a more detailed treatment of the oxidation/reduction (charging) reaction of an electroactive polymer coating. In particular the effects of a changing Donnan potential and site-site interactions are considered. The polymer is assumed to be in quasi-equilibrium with the electrode, as in Section 3. The redox sites react similarly to a redox species in an electrolyte that is confined to a thin layer cell. The concentrations (activities) of the electroactive sites are described by the Nemst equation, cf. Eqn. 29. The current peaks associated with linear sweeps of the electrode potential are well described in the literature see for instance Ref. 62. We can also characterize such systems by charging curves... 

The explanation of the non-linear growth in multilayers as the result of the interdiffusion of the polyelectrolyte chains is based on the existence of a Donnan potential in the polymeric matrix due to the excess of charge due to the adsorption of the layers. This leads to an in and out interdiffusion of at least one type of... 

The polyion domain volume can be computed by use of the acid-dissociation equilibria of weak-acid polyelectrolyte and the multivalent metal ion binding equilibria of strong-acid polyelectrolyte, both in the presence of an excess of Na salt. The volume computed is primarily related to the solvent uptake of tighdy cross-linked polyion gel. In contrast to the polyion gel systems, the boundary between the polyion domain and bulk solution is not directly accessible in the case of water-soluble linear polyelectrolyte systems. Electroneutrality is not achieved in the linear polyion systems. A fraction of the counterions trapped by the electrostatic potential formed in the vicinity of the polymer skeleton escapes at the interface due to thermal motion. The fraction of the counterion release to the bulk solution is equatable to the practical osmotic coefficient, and has been used to account for such loss in the evaluation of the Donnan phase volume in the case of linear polyion systems. 

Chemical potentials inside the membrane are determined by the Flory-Rehner-Donnan equation of state for one-dimensional swelling (5), since the clamped membrane s area. A, is constant. The polymer-solvent interaction parameter, f, is a linear function of the polymer volume fraction, i.e. J + Xi as proposed by Hirotsu (/ 7). Transport of water into and out of the hydrogel membrane causes its thickness, I, to change at a rate that is proportional to the difference in chemical potential of water between the membrane and Cell I and Cell II. 

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