Poisson-Boltzmann equation assumptions

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

What are the assumptions that are needed to obtain the linearized Poisson-Boltzmann (LPB) equation from the Poisson-Boltzmann equation, and under what conditions would you expect the LPB equation to be sufficiently accurate What is the relation between the Debye-Huckel approximation and the LPB equation ... 

The second question concerns one particular aspect of general applicability of the simple mean field equations outlined above as opposed to more sophisticated statistical mechanical descriptions. In particular, the equilibrium Poisson-Boltzmann equation (1.24) is often used in treatments of some very short-scale phenomena, e.g., in the theory of polyelectrolytes, with a typical length scale below a few tens of angstroms (1A = 10-8 cm). On the other hand, the Poisson-Boltzmann equation implicitly relies on the assumption of a pointlike ion. Thus a natural question to ask is whether (1.24) could be generalized in a simple manner so as to account for a finite ionic size. The answer to this question is positive, with several mean field modifications of the Poisson-Boltzmann equation to be found in [5], [6] and references therein. Another ultimately simple naive recipe is outlined below. 

Existence and uniqueness of solutions to the b.v.p. analogous to (2.2.1) has been proved in numerous contexts (see, e.g., [2]—[6]) and can be easily inferred for (2.2.1). We shall not do it here. Instead we shall assume the existence and uniqueness for (2.2.1) and similar formulations and, based on this assumption, we shall discuss some simple properties of the appropriate solutions. These properties will follow from those of the solution of the one-dimensional Poisson-Boltzmann equation, combined with two elementary comparison theorems for the nonlinear Poisson equation. These theorems follow from the Green s function representation for the solution of the nonlinear Poisson equation with a monotonic right-hand side (or from the maximum principle arguments [20]) and may be stated as follows. 

In many practical cases we can use the low-potential-assumption and it leads to realistic results. In addition, it is a simple equation and dependencies like the one on the salt concentration can easily be seen. In some cases, however, we have high potentials and we cannot linearize the Poisson-Boltzmann equation. Now we treat the general solution of the onedimensional Poisson-Boltzmann equation and drop the assumption of low potentials. It is convenient to solve the equation with the dimensionless potential y = ertp/kBT. Please do not mix this up with the spacial coordinate y In this section we use the symbol y for the... 

Fig. 5.1. Reaction rate for the complexation of Ni2 + and PADA (Pyridine-2-azo-p-dimethyl-aniline) in sodium decylsulfate (NaDeS) solutions. — calculated rate from the Poisson-Boltzmann equation. - calculated rate using the assumption that all Ni2+ ions are bound to the micellar surface. (From Ref.285 )...

In a typical macroscopic assumption of proportionality between polarization and applied electric field, P = e0(c — 1 )E, where e is the dielectric constant, and eq3 reduces to the traditional Poisson—Boltzmann equation (the concentrations cH and c0h being in general much smaller than ce). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the macroscopic electric field is obtained7... 

Debye-Huckel approximation — In calculating the potential distribution around a charge in a solution of a strong -> electrolyte, - Debye and -> Hiickel made the assumption that the electrical energy is small compared to the thermal energy ( zjei (kT), and they solved the -> Poisson-Boltzmann equation V2f = - jT- gc° eexp( y) by expanding the exponential... 

Figure 10.3 plots the force between two identical charged plates for different boundary conditions and different assumptions for the potential distribution. These force predictions have been experimentally verified by Pasahley and Israelachvili [19] as shown in Figure 10.4 for the nonlinear Poisson—Boltzmann equation with constant potential boundary conditions. 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

Veo at a large distance from the wall (see Figure 5.67). The magnitude of this electroosmotic velocity was calculated by von Smoluchowski under the assumptions that (1) the ion distribution in the diffuse layer obeys the Poisson-Boltzmann equation, (2) at each point the electrical force is balanced by the viscous friction, and (3) the liquid viscosity in the diffuse layer is equal to that of the bulk liquid, ii. The final result reads ... 

An alternative approach is to prepare a reactive-ion surfactant for which the counterion is itself the reactant and inert counterions and interionic competition are absent (21-23). In principle, this method simplifies estimation of the concentration of an ionic nucleophile, for example, in the micellar pseudophase. Both these treatments of ionic reactions involve assumptions and approximations that seem to be satisfactory, provided that ionic concentrations are low, e.g., <0.1 M. These assumptions and approximations fail when electrolyte concentrations are high (24-25). A more rigorous treatment is based on application of the Poisson-Boltzmann equation in spherical symmetry (26-28), and this treatment accounts for some of the failures of the simpler models (29, 30). 

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... 

Besides the hypothesis that the Poisson equation (Eq. 3.26), from which Eq. 5.1 is derived, is physically meaningfol when x is measured over molecular dimensions, there are four basic assumptions embodied in the Poisson-Boltzmann equation as written above ... 

The limitations imposed on DDL theory as a molecular model by these four basic assumptions have been discussed frequently and remain the subject of current research.In Secs. 1.4 and 3.4 it is shown that DDL theory provides a useful framework in which to interpret negative adsorption and electrokinetic experiments on soil clay particles. This fact suggests that the several differences between DDL theory and an exact statistical mechanical description of the behavior of ion swarms near soil particle surfaces must compensate one another in some way, at least in certain applications. Evidence supporting this conclusion is considered at the end of the present section, whose principal objective is to trace out the broad implications of Eq. 5.1 as a theory of the interfacial region. The approach taken serves to develop an appreciation of the limitations of DDL theory that emerge from the mathematical structure of the Poisson-Boltzmann equation and from the requirement that its solutions be self-consistent in their physical interpretation. TTie limitations of DDL theory presented in this way lead naturally to the concept of surface complexation. 

The second method is to neglect the convection term in Eq. 9 considering that the flow velocity is small for both EOF flow and electrophoretic flow, which decouples the flow field and the EDL potential field. In the second approach, the assumption of a two-species buffer and the application of the Boltzmann distribution are commonly made in order to solve the potential field easily, which yields the well-known Poisson-Boltzmann equation ... 

In the small nanochaimels (from a few to about 100 nm), the electric double layer (EDL) thickness becomes larger or at least comparable with the nanochaimels lateral dimensions. It affects the balance of bulk ionic concentrations of co-ions and counterions in the nanochannels. Thus, many conventional approaches such as the Poisson—Boltzmann equation and the Helmholtz-Smoluchowski slip velocity, which are based on the thin EDL assumption and equal number of co-ions and counterions, lose their credibility and cannot be utilized to model the electrokinetic effects through these nanoscale channels. The Poisson equation, the Navier-Stokes equations, and the Nemst-Planck equation should be solved directly to model the electrokinetic effects and find the electric... 

It is assumed that the electric charge density is not affected by the external electric fields due to the thin EDLs and small fluid velocity therefore, the charge convection can be ignored, and the electric field equation and the fluid flow equation are decoupled. Based on the assumption of local thermodynamic equihbrium, for small zeta potential, the electric potential due to the charged wall is described by the linear Poisson-Boltzmann equation which can be written in terms of dimensionless variables as... 

Abstract The electric fields and potential in a pore filled with water are calculated, without using the Poisson-Boltzmann equation. No assumption of macroscopic dielectric behavior is made for the interior of the pore. The field and potential at any position in the pore are calculated for a charge in any other position in the pore, or the dielectric boundary of the pore. The water, represented by the polarizable PSPC model, is then placed in the pore, using a Monte Carlo simulation to obtain an equilibrium distribution. The water, charges, and dielectric boundary, together determine the field and potential distribution in the channel. The effect on an ion in the channel is then dependent on both the field, and the position and orientation of the water. The channel can exist in two major configurations open or closed, in which the open channel allows ions to pass. In addition, there may be intermediate states. The channel has a water filled pore, and a wall... 

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