Cylindrical cell model

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. 

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...

Deserno M, Holm C, Kremer K (2000) Molecular dynamics simulations of the cylindrical cell model. In Radeva T (ed) Physical Chemistry of Polyelectrolytes. Marcel Dekker,... 

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...

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 common further approximation assumes that the investigation of a small subvolume containing only one rod and its counterions will suffice to unveil much of the interesting physics. The main justification for this approach is that the subvolume has zero net charge. Moreover, the counterions will also efficiently screen higher order multipoles. Hence the interactions between two such subvolumes, which are neglected when focusing on just one rod, will be fairly weak. This approximation is called the cylindrical cell model, and it provides the framework for this study. 

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-... 

Tal, R., Lee, D., and Siriganano, W. Hydrodynamics and heat transfer in sphere assemblages cylindrical cell models. Int. J. Heat Mass Transf. 26(9), 1265-1273, 1983. 

Tal, R. and Sirignano, W. Cylindrical cell model for hydrodynamics of particles assemblages at intermediate Reynolds numbers. AIChE J. 28(2), 233-237, 1982. 

Fig. 5 a Mean force and b potential of mean force for Systems I-IV as a function of the macroion separation evaluated from the cylindrical cell model containing two macroions and 22r counterions. In b, = 5J5Rm) = 0 has been adopted. Rcy = 4J m> f cyl =... 

Van, H., and J. A. M. Smit. 1995. Approximative analytical solutions of the Poisson-Boltzmann equation for charged rods in the presence of salt An analysis of the cylindrical cell model. Journal of Colloid and Interface Science 170, no. 1 134—145. doi 10.1006/ jcis.1995.1081. 

In dilute polyelectrolyte solutions without added salt, the Poisson-Boltzmann cylindrical cell model accounts fairly well for thermodynamic and some transport properties observed [110-112]. Accordingly, the osmotic pressure in such solutions may be expressed in a virial expansion as commonly used with only two terms [110] ... 

Hence, the experimental data are consistent with the charged cylindrical cell model. The chain thus appears to be stretched on the length scale of the few nanometers probed by the ESR experiments. 

As the water-ethanol mixture has lower permittivity ( = 50) than water (e 80), while the water-N-methylpropionamide mixture has higher permittivity (e 140), it is somewhat surprising that ESR spectra for FS counterions in an aqueous solution of PDADMAC are not consistent with the charged cylindrical cell model. The most likely cause for the failure of the model for pure water are differences in solvation of the polyelectrolyte chain compared to the mixed solvents. The organic components of both solvent mixtures feature ethyl groups that are better suited than water for solvating the hydrophobic parts of the PDADMAC chains. They may thus prevent a local hydrophobic collapse of the chain. 

Ion concentrations at the cell boundary are found from the analogue of Eq. [246] with the osmotic pressure n given by the middle expression in Eq. [247]. Unfortunately, no exact analytical solution has been found for the important case of the cylindrical cell model with larger amounts of added salt, but good approximations are available. ... 

H. Van Keulen and J. A. M. Smit,/. Colloid Interface Sci., 170, 134 (1995). Approximate Analytical Solutions of the Poisson-Boltzmann Equation for Charged Rods in the Presence of Salt An Analysis of the Cylindrical Cell Model. 

M. Desemo, C. Holm, and K. Kremer, in Physical Chemistry of Polyelectrolytes, T. Radeva, Ed., Marcel Dekker, New York, 2001, pp. 59-110. Molecular Dynamics Simulations of the Cylindrical Cell Model. 

The measmed line shapes were clearly inconsistent with the RDC PMo(r) = exp(-fer) predicted by the simple Marming-Oosawa theory of counterion condensation. Almost perfect agreement was obtained for solutions of PDADMAC in mixtures of water with ethanol (50vol.%, meanpermittivity 50) or N-methylpropionamide (NMPA, 77 vol.%, =140) for the charged cylindrical cell model, which predicts Pcccm (r) =Po r " (see, for example, fit in Figure 16(c)). This model is applicable to semidilute solutions of polyelectrolytes where... 

---