Macroion, defined

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

The situation illustrated in Figure 4. la has by now become familiar. It depicts a gel composed of a parallel stack of plate macroions with a well-defined interplate spacing (in the colloidal range 10 to 100 nm) in equilibrium with a supernatant fluid. Let us think of the boundary of the gel as an effective membrane enclosing the macroions, transforming the picture into Figure 4.1b. This chapter is concerned with the calculation of the distribution of salt between the gel (I) and supernatant fluid (II) in the two-phase region of colloid stability. 

As g can also be calculated in terms of the surface potential (see below), Nlic can be calculated as a function of 4>s. When Ade is known, the average number density of the deficit, nie, can be calculated from Equation 4.6 the position of the coulombic attraction theory minimum is connected to the average electrolyte concentration in the gel phase because it defines the average volume occupied per macroion. The number density of the deficit of negative ions in the gel phase is given by... 

Let us recall the schematic illustration of the raw phenomenon of the clay swelling in Figure 1.4. In the cases studied in Chapters 1 to 3, V was always much greater than V, the volume occupied by the macroions. We now define Vm to be the volume occupied by the macroions in the coagulated (crystalline) state, as in Figure 1.4a in the vermiculite system. This is an experimentally controlled variable. We define the sol concentration r by... 

It is a misinterpretation of the concept of a phase to consider these two possible sites for a counterion as separate phases. This confuses a macroscopic property of a well-defined phase (a solid) in equilibrium with another well-defined phase (a homogeneous electrolyte solution) with an internal property of an inhomogeneous single phase, the macroionic (or colloidal or gel) phase. It divides the macroionic phase into two regions that have no physical counterpart. 

This results in each surface layer consisting of a well-defined sublayers in an immediate vicinity of the film surface the shape of the density profiles of the surface sublayers has a 5-like form indicating that surface sublayers are the quasi-two-dimensional monolayers. The surface layers formed within the DLVO-like model being thinner than the middle-film layers still are far from to be monolayers. As a result, the segregation of the middle-film layers from the surface layers is not so evident in this case. As expected, the difference between models with and without excluded volume forces increases when the macroion charge becomes smaller (Fig. 9b). 

A further increase of the wedge thickness (the MC data are not presented here) is accompanied by the transformation of the macroion film (in between two wedge-shaped surface monolayers) from bilayer to threelayer, from three-layer to fourlayer and so on, with the each new layer being less and less defined up to approaching the structuring similar to that in a bulk solution. 

The main results presented in this chapter are obtained for macroion suspension defined by the bulk volume fraction of the macroion particles, r) = 0.05, the macroion charge number, Z = 30, the Debye length of the supporting electrolyte, i I) - 2. Such a choice of the parameters makes this model system a suitable prototype of some aqueous ionic micellar solutions such as cetyltrimethylammonium bromide (CTAB) and sodium dodecyl sulfate anionic surfactant (SDS). 

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. 

In poor solvent, the size R of the macroion in both cases is defined by non-Coulomb interaction ... 

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