Poisson-Boltzmann cell model

The fact that Poisson-Boltzmann theory overestimates the osmotic coefficient is well-known in literature. Careful studies of typical flexible polyelectrolytes in solution ([2, 23] and further references given there) indicated that agreement of the Poisson-Boltzmann cell model and experimental data could only be achieved if the charge parameter was renormalized to a higher value. To justify this procedure it was assumed that the flexible polyelectrolytes adopt a locally helical or wiggly main chain in solution. Hence,... 

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. 

Blaul, J., Wittemann, M., Ballauff, M., and Rehahn, M. Osmotic coefficient of a synthetic rodlike polyelectrolyte in salt-free solution as a test of the Poisson-Boltzmann cell model. Journal of Physical Chemistry B, 2000,104, No. 30, p. 7077-7081. 

Here we review the application of ASAXS as applied to the analysis of stiff chain polyelectrolyte in solution. The data discussed here [19] have been obtained using the polyelectrolyte the chemical structure of which is shown in Fig. 1. This system has already been under scrutiny by conventional SAXS some time ago [14]. The paper is organized as follows first we summarize the theory of ASAXS and its application to the problem at hand [18]. Moreover, we will briefly summarize the treatment of rod-like polyelectrolytes within the frame of the Poisson-Boltzmann cell model. An important point for the present analysis is the influence of mutual interaction of the dissolved polyelectrolytes. ASAXS-measurements need to be done at rather higher concentrations so that the interaction of the solute rods may come into play. Here it will be shown that this problem is negligible for the present system. Next possible difficulties encountered in an ASAXS experiment will be discussed and experimental results will be presented. A brief final section will conclude the present discussion. 

Here Ro denotes the cell radius introduced by the Poisson-Boltzmann cell model (see below). The term Ap rc) may therefore be split into a non-resonant contribution Apo(rc) and the resonant contributions of the counterions according to [17],... 

Comparison with the Poisson-Boltzmann cell model... 

SAXS and osmometry, on the other hand, allow the conclusion that the Poisson-Boltzmann cell model gives a quite realistic description of counterion condensation in rodlike macromolecules. However, prior to a final evaluation, a more profound analysis is required. Here, it will be of particular importance to consider polyelectrolytes with substantially lower charge densities also. Unfortunately, but in accordance with expectations, all polyelectrolytes containing phenylene moieties without charged side groups, such as 20-22, proved to be insoluble in water (Scheme 4). 

Fig. 8 Poisson-Boltzmann cell model (a) different ways to model a polyelectrolyte chain by a cylindrical rod, (b) different ways to choose the cell boundaries...

Biesheuvel PM, Lindhoud S, de Vries R et al (2006) Phase behavior of mixtures of oppositely charged nanoparticles heterogeneous Poisson-Boltzmann cell model applied to lysozyme and succinylated lysozyme. Langmuir 22 1291-1300... 

Together these essentially replace the Poisson-Boltzmann cell model with the Debye-Hiickel bulk model, allowing many more systems to be treated analytically, although not necessarily accurately, and providing considerable insight into the physical characteristics of electrolyte solutions. 

J. Blaul, M. Witteman, M. Ballauff, and M. Rehahn, /. Phys. Chem. B, 104, 7077 (2000). Osmotic Coefficient of a Synthetic Rodhke Polyelectrolyte in Salt-Free Solution as a Test of the Poisson-Boltzmann Cell Model. 

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

Poisson-Boltzmann Theory for the Cylindrical Cell Model. 5... 

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. 

Deserno M, Holm C (2001) Cell model and Poisson-Boltzmann theory a brief introduction. In Holm C, Kekicheff P, Podgornik R (eds) Electrostatic Effects in Soft Matter and Biophysics. Kluwer, Dordrecht, p 27... 

An alternative theoretical approach is the application of the Poisson-Boltzmann equation on the so-called cell model, assuming a parallel and equally spaced packing of rod-like polyions [62, 63]. This allows one to calculate at finite concentration according to ... 

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... 

Fig. 1 Counterion distribution function P(r) from Eq. (1) for two cylindrical cell models with R/b= 123.8,1=0.959 e0/b and the values for Bjerrum length and valence as indicated in the plots. The solid line is the result of a molecular dynamics simulation [9] while the dotted line is the prediction from Poisson-Boltzmann theory. The increased counterion condensation visible in the simulation is accurately captured by the extended Poisson-Boltzmann theory (dashed line) using the DHHC correction from Ref. [18]. An approach using the DHH correction from Ref. [16] (dash-dotted line) evidently fails to correctly describe the ion distribution...

Wennerstrom, H., Jonsson, B., and Linse, P. The cell model for poly-electrolyte systems - exact statistical mechanical relations, Monte-Carlo simulations, and the Poisson-Boltzmann approximation. Journal of Chemical Physics, 1982, 76, No. 9, p. 4665 -670. 

Das, T., Bratko, D., Bhuiyan, L.B., and Outhwaite, C.W. Polyelectrolyte solutions containing mixed valency ions in the cell model A simulation and modified Poisson-Boltzmann study. Journal of Chemical Physics, 1997,107, No. 21, p. 9197-9207. 

Keywords cell signaling lipid rafts BAR domains membrane curvature membrane elasticity PIP2 diffusion mean-field model coarse-grained theory Poisson-Boltzmann theory Cahn-Hilliard equations... 

Within the anisotropic cylindrical cell model the pressure is related to volume changes leaving the direction along the rod invariant. Simulations constantly yield a smaller osmotic coefficient than predicted by Poisson-Boltzmann theory. For multivalent systems it can even become negative. 

A rational description of ionic atmosphere binding is provided by the Poisson-Boltzmann equation and the cylindrical cell model. Figure 1 is an example of such computations and shows the variation of the local concen-... 

The emphasis placed on the last assumption is responsible for the name of the model. It is now well known that these assumptions, especially the first two, are reliable with impunity only over very narrow and dilute micellar concentration ranges. Nevertheless, the PIE model has provided invaluable insight over the past 25 years in elucidating micellar catalysis. Its failures [27-31] are usually attributable to clear-cut violations of its simple assumptions. Refinements or alternatives to these basic premises such as solving the nonlinear Poisson Boltzmann equation for the cell model have not proved to be particularly enlightening nor more helpful [32]. The extension of the PIE model to complicated micellar systems where anomalous rate behavior is more often than not the rule rather than the exception is probably unwarranted [33]. Sudhdlter et al. [34] have critically reviewed the Berezin model and its Romsted variation, the PIE model, as matters stood 20 years ago. In... 

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