Cell model theories

Taken together, the static lattice and cell model theories strongly suggest that ... 

Before presenting numerical results, it is worth summarizing the main characteristics of the experimental results for the osmotic pressure of polyelectrolyte solutions [9, 17, 18, 57, 107], The measured osmotic coefficients most often exhibit strong negative deviations from ideality. The measured values are a) lower than it was predicted by the cylindrical cell model theory, b) rather (but not completely) insensitive to the nature of the counterions, and c) also insensitive to the polyelectrolyte concentration in a dilute regime and/or for... 

A. The cell model theoiy. The cell model theory was first introduced by Lennard-Jones and Devonshire to calculate the bulk properties of molecular fluids (Barker, 1%3). In this approach, the array of particles is replaced by an array of hypothetical cells inside which the movement of each particle is confined. 

The Monte Carlo calculations provide a yardstick against which the results of the cell model theory is compared. The agreement between the two methods is, in general, excellent. 

Saxton JA, Fitton JB, Vermeulen T. Cell model theory of homogeneous fluidization density and viscosity behaviour. AIChE J 16 120-130, 1970. 

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. ... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

Concerning point (b), a generalized theory was developed independently by Prigogine and his co-workers10 11 and by Scott.18 The main idea was to combine the concept of average potential involved in the cell model with the theorem of corresponding states for pure compounds, in such a way that ... 

This equation is predicted by the mixing cell model, and turbulence theories put forward by Aris and Amundson130 and by Prausnttz(31). 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

The beginning of modeling of polymer-electrolyte fuel cells can actually be traced back to phosphoric-acid fuel cells. These systems are very similar in terms of their porous-electrode nature, with only the electrolyte being different, namely, a liquid. Giner and Hunter and Cutlip and co-workers proposed the first such models. These models account for diffusion and reaction in the gas-diffusion electrodes. These processes were also examined later with porous-electrode theory. While the phosphoric-acid fuel-cell models became more refined, polymer-electrolyte-membrane fuel cells began getting much more attention, especially experimentally. 

In the dense system the convergence of the virtual expansion is doubtful. In any case the higher coefficients are hard lo calculate here approximate theories have been developed, of which the cell model is an example,... 

S. Ramasesha, Z. G. Soos, in Valence Bond Theory, D. L. Cooper, Ed., Elsevier, Amsterdam, The Netherlands, 2002, pp. 635-697. Valence Bond Theory of Quantum Cell Models. 

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. 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

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. 

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. 

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]. 

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]. 

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...

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,... 

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. 

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... 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

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... 

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