PB-cell model

Fig. 1 Scheme of the PB cell model The rod-like macroion with radius a is confined in a cell of radius R0 together with its counterions. The charge density of the macroion is characterized by the charge parameter (see Eq. (1)). See Sect. 2.1 for further explanation... 

Another viable method to compare experiments and theories are simulations of either the cell model with one or more infinite rods present or to take a solution of finite semi-flexible polyelectrolytes. These will of course capture all correlations and ionic finite size effects on the basis of the RPM, and are therefore a good method to check how far simple potentials will suffice to reproduce experimental results. In Sect. 4.2, we shall in particular compare simulations and results obtained with the DHHC local density functional theory to osmotic pressure data. This comparison will demonstrate to what extent the PB cell model, and furthermore the whole coarse grained RPM approach can be expected to hold, and on which level one starts to see solvation effects and other molecular details present under experimental conditions. 

Up to now, only two sets of data of the osmotic coefficient of rod-like polyelectrolytes in salt-free solution are available 1) Measurements by Auer and Alexandrowicz [68] on aqueous DNA-solutions, and 2) Measurements of polyelectrolyte PPP-1 in aqueous solution [58]. A critical comparison of these data with the PB-cell model and the theories delineated in Sect. 2.2 has been given recently [59]. Here it suffices to discuss the main results of this analysis displayed in Fig. 8. It should be noted that the measurements by the electric birefringence discussed in Sect. 4.1 are the most important prerequisite of this analysis. These data have shown that PPP-1 form a molecularly disperse solution in water and the analysis can therefore assume single rods dispersed in solution [49]. 

The osmotic coefficient obtained experimentally from polyelectrolyte PPP-1 having monovalent counterions compares favorably with the prediction of the PB cell model [58]. The residual differences can be explained only partially by the shortcomings of the PB-theory but must back also to specific interactions between the macroions and the counterions [59]. SAXS and ASAXS applied to PPP-2 demonstrate that the radial distribution n(r) of the cell model provides a sufficiently good description of experimental data. 

Evidently, more work has to be done for a comprehensive comparison of theory and experiment. Theory and simulations reveal clearly that the PB-cell model should be a poor approximation for divalent counterions and breaks down totally for trivalent counterions [29]. A comprehensive experimental test of these very important conclusions is still missing. 

Figure 6. MC results for multicomponent model (filled circles) and PB cell model results (open circles connected with line) for the ratio k/k° at zp = — 60 and for Ce = 0.005 mol dm 3 as a function of the macroion concentration.

So the results were compared to the PB cell model calculation (cf Sec. 2). Considering that the PB theory treats the counterions and co-ions as a cloud of electrical charge around the macroion, the k/k° was estimated by using the equation of Morawetz [77, 78],... 

An extensive discussion of the PB cell model has recently been presented by Deserno and coworkers [8] (see also preceding chapter by Vlachy et al). Hence, it suffices here to delineate the main features and how this model is compared to data deriving from a scattering experiment. The PB cell model treats the system as an assembly of N rods confined in cells of radius Rq. ... 

As mentioned above, ASAXS measurements require rather high concentrations of the dissolved polyelectrolytes. The cell model assumes a decoupling of the different polyelectrolytes. All calculations discussed in the Sec. 2.2 are referring explicitly to a hypothetical state in which the concentration of the polyelectrolyte is finite, i. e., the cell radius R<> assumes a finite value but S(q) (see Eq. (3)) is unity. Considering the influence of finite concentration it is important to note that the PB cell model predicts that most of the counterions are located in the immediate vicinity of the macroion (see the model calculations in [18]. Hence, an increase of concentration followed by a decrease of the distance between the rods is hardly seen in Io(q) calculated by this model. For the range of concentrations that are ranging from 1 to 20.0 g/L the cloud of counterions therefore does not change its spatial distribution in a profound manner. This prediction which is borne out of the cell model is well confirmed by experiments [18]. 

For 5 = 0, we need only take the limit 8 —> 0 in Eq. [238] for 8 < 0 (the low surface charge density case), we put 8 —> i8 and by analytic continuation obtain the tanh function. The form of the integral chosen is appropriate for highly charged cylindrical polyelectrolytes such as DNA. We then have the solution for the RMP within the PB cell model with no added salt... 

Figure 23 The radial Manning parameter (r) (top frame, cell model only Eq. [244]) and potential c )(r) (bottom frame Eq. [245]) in the PB cell and bulk models for a charged cylinder of radius <3 = 10 A and surface charge density = —0.094 e(jk (corresponding to an average charge spacing of h = 1.69 A as in B-DNA). A site concentration (corresponding to a phosphate concentration in DNA) of 0.1 M has been chosen, giving a Manning radius of Rm = 29.6 A and a cell radius of R = 56 A. The PB cell model potential profile (solid lines) is compared to the bulk PB (dotted-dashed line Eq. [389]), DH cell model (dashed lines Eq. [256]), DH bulk model (dotted line Eq. [259]), and no-ion (circles Eq. [362]) values.

The potential can be obtained from the (r) profile through a simple integration (Eq. [230]). The difference in (r)/r between the DH and PB values is also shown in Figure 23 (top frame, dotted line) with most of the difference occurring within one Debye length of the surface = 13.6 A). Of particular relevance to biophysical systems is the competition between mono- and divalent counterions at the cylindrical surface,the discussion of which we defer until later, and that between monovalent counterions with different radii. Also, Deserno and Holm have compared molecular dynamics simulations with the prediction of PB cell model theory for the calculation of osmotic coefficients and the quantification of counterion condensation. ... 

Within the PB cell model, the counterion concentration at the outer boundary gives the osmotic pressure, which is a measure of the electrostatic repulsion between neighboring biomolecules. This pressure can also be experimentally determined. " " The Donnan coefficient, on the other hand, is strongly influenced by conditions at the macromolecular surface and can be used to provide key insight into the nature of polyelectrolyte-counterion interaction. " This interaction is important because of the salt-induced conformational changes DNA undergoes. The nature of this behavior is believed to arise from the partial collapse, or condensation. 

As with the analytical solutions presented above, we consider in detail the PB cell model and take the bulk model limit at the end. We begin the derivation of the finite-element PB algorithm by writing the mobile charge density of Eq. [4] as an explicit functional of the potential ... 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

As already indicated in Sect. 2, the osmotic coefficient 0 provides a sensitive test for the various models describing the electrostatic interaction of the counterions with the rod-like macroion. It is therefore interesting to first compare the PB theory to simulations of the RPM cell model [26, 29] in order to gain a qualitative understanding of the possible failures of the PB theory. In a second step we compare the first experimental values 0 obtained on polyelectrolyte PPP-1 [58] quantitatively to PB theory and simulations [59]. 

Fig. 8 Osmotic coefficient as a function of counterion concentration cc for the poly(p-phenylene) systems described in the text. The solid line is the PB prediction of the cylindrical cell-model, the dashed curve is the prediction from the correlation corrected PB theory from Ref. [58]. The full dots are experiments with iodine counterions and the empty dots are results of MD simulations described in ref. [29,59]. The Manning limiting value of l/2 is also indicated...

A survey over the area of stiff-chain polyelectrolytes has been given. Such rod-like polyelectrolytes can be realized by use of the poly(p-phenylene) backbone [9-13]. The PPP-polyelectrolytes present stable systems that can be studied under a wide variety of conditions. Moreover, electric birefringence demonstrates that these macroions form molecularly disperse solution in water [49]. The rod-like conformation of these macroions allows the direct comparison with the predictions of the Poisson-Boltzmann cell model [27-30] which has been shown to be a rather good approximation for monovalent counterions but which becomes an increasingly poor approximation for higher valent counterions [29]. Here it was shown in Sect. 2.2 that the basic problem of the PB model, namely the neglect of correlations, can be remedied in a systematic fashion. 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

For high rod charge, i.e., large , this expression becomes independent of valence. Hence, replacing monovalent counterions by multivalent ones reduces their total number in the cell, but for a highly charged rod not their density at the rod surface. This statement is in fact an exact result for the cell model, and not a consequence of the PB approximation [27],... 

For an isolated cell the pressure is given by the particle density at the outer cell boundary. As a corollary, it must then also be positive. This is a rigorous statement, true for the spherical, cylindrical, and planar cell model [27], It is a merit of the PB equation that it retains the validity of this exact relation. For the extended density functional theories this no longer holds, and additional terms appear [5]. 

Up to now the following observation repeatedly turned up from the simulations. The nonlinear PB equation provides a fairly good description of the cell model, but it suffers from systematic deviations in strongly coupled or dense systems. It underestimates the extent of counterion condensation and at the same time overestimates the osmotic coefficient. As the common reason for both problems, the neglect of correlations has been proposed, basically for two reasons ... 

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