Poisson-Boltzmann, approach

Finally, surprisingly good initial results have been obtained with a simple approach that makes use of the Poisson-Boltzmann approach to estimating solvation free energies. In this method,74"76... 

Other deviations and limitations are expected from the use of the Poisson-Boltzmann approach, two of which are... 

Liu, H. Y., and Zou, X. (2006). Electrostatics of ligand binding Parametrization of the generalized Bom model and comparison with the Poisson-Boltzmann approach. J. Phys. Chem. B 110, 9304-9313. 

A long time ago, Stem noted that the traditional assumption that the ions interact only with a mean electrical field (the Poisson - Boltzmann approach) leads to an ionic density in the vicinity of the interface that exceeds the available volume. A simple way to avoid this difficulty is to consider that the ions are hydrated, and therefore there are fewer positions available to them in the vicinity of charged surfaces [4.1]. When compared to the traditional Poisson-Boltzmann result, this correction leads to an increase in the repulsive force at... 

The mean-field formalism (the Poisson-Boltzmann approach). 65... 

The Poisson-Boltzmann approach [1,2] has the advantage of simplicity and is surprisingly accurate, at least for univalent ions in a certain range of electrolyte concentrations (1.0X10 3-5X10 2 M) and not too close to the interface. It was later employed [10] to explain the repulsion due to the overlap of two double layers, and the stability of colloids thereafter [11,12],... 

Another limitation of the Poisson-Boltzmann approach is that the interaction between two surfaces immersed in water might not be exclusively due to the electrolyte ions. For instance, water has a different structure in the vicinity of the surface than in the bulk and the overlapping of such structures generates a repulsion even in the absence of electrolyte [20]. In this traditional picture, the hydration repulsion is not related to ion hydration actually it is not related at all to electrolyte ions. However, as recently suggested [21], this hydration interaction can still be accounted for within the Poisson-Boltzmann framework, assuming that the polarization is not proportional to the macroscopic electric field, but depends also on the field generated by the neighboring water dipoles and by the surface dipoles. 

Fig. 8. The change in surface tension at low electrolyte concentrations (the Jones-Ray effect). Circles Experimental data from Ref. [33]. Curve (1) predicted surface tension for the simple model which accounts for OH adsorption and only ion hydration effects for electrolyte ions, within the Poisson-Boltzmann approach. Curve (2) predicted surface tension when image forces are also included in the model. As noted in Section 3, the image forces cannot be neglected at concentration lower than 0.01 M.

The limitations of the traditional Poisson-Boltzmann treatment are numerous. A particularly important one refers to specific ion effects. The amphoteric latex particles coagulate when the concentration of CsN03 is approximately 1 M, but remain stable at high pH values even when the concentration of KN03 exceeds 3 M [6]. Nevertheless, in the traditional Poisson-Boltzmann approach the double layer repulsions for Cs and K are the same. A number of attempts were made to include additional interactions in the formalism, some of them being briefly discussed in the first part of this article [7], With these corrections, many of them dependent on unknown parameters, the modified Poisson-Boltzmann approach regained its explanatory power, and in general most of the experiments could be accounted for... 

In the inset of Fig. 6c, the calculations were repeated for the same values of the parameters, but a stronger double layer (N= 2X1017 sites/m2, Ku = 1.0 M). The ion-dispersion forces have in this case only a minor effect, and the interaction can be well approximated by the traditional Poisson-Boltzmann approach, with slightly modified parameters (density of sites, dissociation constant). 

The situation is similar to the success of the traditional Poisson—Boltzmann approach its ability in describing, at least qualitatively, and many times even quantitatively, the behavior of most colloidal systems probably resides in the use of at least one adjustable parameter (surface charge, surface potential, recombination constant and so on) in the fitting of the experimental results. If that parameter could be accurately measured, one would have to address the inaccuracies generated by the mean field treatment itself. 

Whereas the corrections to the traditional Poisson— Boltzmann approach could explain many experimental results, there are systems, such as the vesicles formed by neutral lipid bilayers in water, for which an additional force is required to explain their stability.4 This force was related to the organization of water in the vicinity of hydrophilic surfaces therefore it was called hydration force .5... 

Another clear failure of the Poisson-Boltzmann approach was provided by the experiments regarding the force between neutral lipid bi layers [11], The repulsion required to explain their stability was determined to have an almost exponential dependence, with a decay length of about 2—3 A [11], and neither this decay length nor the magnitude of the interactions were dependent on the ionic strength. This interaction was initially attributed to the structuring of water near the surface (the hydration of the surfaces) and it is usually called hydration force [12]. The microscopic origins of this interaction are still under debate. 

Most of the water-mediated interactions between surfaces are described in terms of the DLVO theory [1,2]. When a surface is immersed in water containing an electrolyte, a cloud of ions can be formed around it, and if two such surfaces approach each other, the overlap of the ionic clouds generates repulsive interactions. In the traditional Poisson-Boltzmann approach, the ions are assumed to obey Boltzmannian distributions in a mean field potential. In spite of these rather drastic approximations, the Poisson-Boltzmann theory of the double layer interaction, coupled with the van der Waals attractions (the DLVO theory), could explain in most cases, at least qualitatively, and often quantitatively, the colloidal interactions [1,2]. 

Eqs. (1), (3) and (4) represent a Modified Poisson-Boltzmann approach, which, when the ions interacts only via the mean field potential ip(z) (e.g. AWa=AWc=0), reduces to the well-known Poisson-Boltzmann equation. The only change in the polarization model is the replacement of the constitutive equation Eq. (1) by Eq. (2), which accounts for the correlation in the orientation of neighboring dipoles. However, since independent functions, four boundary conditions are needed to solve the system composed of Eqs. (2) (3) and (4). For only one surface immersed in an electrolyte, t]/(z-co)-0, m(z-co)=0 whereas for two identical surfaces, the symmetry of ip and m requires that = 0 and m(z=0)=0 where z- 0 represents... 

In summary, the polarization model represents an extended Poisson-Boltzmann approach, in which the hydration and the double layer are not independent interactions, but are intimately coupled to each other, via an electrostatic coupling between the fields ip(z) and m(z). These fields can be calculated by solving Eqs. (2) (3) and (4), and the total free energy of the (5b) system can be obtained by summing up the terms provided by... 

Eqs. (5b) (5c) (5d) (5e). The constitutive equation of the polarization model contains only one unknown parameter, namely A, which expresses the correlation between neighboring dipoles. However, in order to solve the system of equations, both the surface charge density a (or the surface potential ip(z —d)) as well as the polarization of water near the surface have to be known. The latter can be related, in a microscopic model that will be examined in the next section, to the surface dipoles. In the limit A,—>0, the polarization model reduces to the Poisson— Boltzmann approach, and the two boundary conditions become dependent on each other, because in this case... 

It is not surprising that the Poisson-Boltzmann approach has been used frequently in computing interactions between charged entities. Mention may be made of the Gouy theory (Fig. 3.24) of the interaction between a charged electrode and the ions in a solution (see Chapter 6). Other examples are the distribution (Fig. 3.25) of electrons or holes inside a semiconductor in the vicinity of the semiconductor-electrolyte interface (see Chapter 6) and the distribution (Fig. 3.26) of charges near a polyelectrolyte molecule or a colloidal particle (see Chapter 6). 

The most expensive part of a simulation of a system with explicit solvent is the computation of the long-range interactions because this scales as Consequently, a model that represents the solvent properties implicitly will considerably reduce the number of degrees of freedom of the system and thus also the computational cost. A variety of implicit water models has been developed for molecular simulations [56-60]. Explicit solvent can be replaced by a dipole-lattice model representation [60] or a continuum Poisson-Boltzmann approach [61], or less accurately, by a generalised Bom (GB) method [62] or semi-empirical model based on solvent accessible surface area [59]. Thermodynamic properties can often be well represented by such models, but dynamic properties suffer from the implicit representation. The molecular nature of the first hydration shell is important for some systems, and consequently, mixed models have been proposed, in which the solute is immersed in an explicit solvent sphere or shell surrounded by an implicit solvent continuum. A boundary potential is added that takes into account the influence of the van der Waals and the electrostatic interactions [63-67]. 

Quesada-Perez, M., Gonzalez-Tovar, E., Martin-Molina, A., Lozada-Cassou, M., and Hidalgo-Alvarez, R. Overcharging in colloids Beyond the Poisson-Boltzmann approach. ChemPhysChem, 2003, 4, No. 3, p. 235-248. 

Counterion condensation theory, however, does not provide a detailed picture of the distribution of the condensed Ions. Recent research using the Poisson-Boltzmann approach has shown that for cylindrical macroions exceeding the critical linear charge density the fraction of the counterions described by Manning theory to be condensed remain within a finite radius of the macroion even at infinite polyion dilution, whereas the remaining counterions will be infinitely dispersed in the same limit. This approach also shows that the concentration of counterions near the surface of the macroion is remarkably high, one molar or more, even at infinite dilution of the macromolecule. In this concentrated ionic milieu specific chemical effects related to the chemical identities of the counterions and the charged sites of the macroion may occur. 

Verwey and Niessen treated the interfacial ion distribution as two back-to-back double layers, each described using the Poisson-Boltzmann approach developed by Gouy and Chapman for electrode-electrolyte interfaces [13]. For a 1 1 electrolyte, the excess ionic charge density on the aqueous side of the interface, qw, is given by... 

Relative to finite-difference Poisson-Boltzmann approaches, such methods have the advantage that only the two-dimensional cavity surface must be discretized. 

Surface potentials at the electrode-solution interface have been described by a number of formalisms. The most successful of these originates from Gouy and Chapman whom suggested that Poisson-Boltzmann approaches best describes the state of affairs at the electrode surface in contact with an aqueous solution (further elaborations are outlined by Bockris Reddy). Within Electrochemistry this proved very successful and analogous formalisms were subsequently applied to physical descriptions of biological surfaces. The resultant Poisson-Boltzmann equation with defined boundary conditions can be solved analytically to yield an expression for the membrane surface potential as follows ... 

Poisson-Boltzmann equation, more sophisticated analytical alternatives, or computer simulation. Both Debye-Hiickel and Poisson-Boltzmann approaches offer a mean-field description of electrostatic screening by small (point-like), thermally equilibrated ions such descriptions fail to address the spatial correlations in placement of these ions that may be important in real polyelectrolyte systems. 

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