Linearization approximation Debye-Hiickel

When the interface is spherical, the electrostatic potential is solved explicitly only for a linearized case (Debye-Hiickel approximation [Eq. (8)], although there have been a number of attempts to obtain the exact sol-... 

Fig. 1.8 Dependence of the mean activity coefficient y tC of NaCl on the square root of molar concentration c at 25°C. Circles are experimental points. Curve 1 was calculated according to the Debye-Hiickel limiting law (1.3.25), curve 2 according to the approximation aB = 1 (Eq. 1.3.32) curve 3 according to the Debye-Hiickel equation (1.3.31), a = 325nm curve 4 according to the Bates-Guggenheim approximation (1.3.33) curve 5 according to the Bates-Guggenheim approximation + linear term 0.1 C curve 6 according to Eq. (1.3.38) for a = 0.4nm, C = 0.055dm5-mor ...

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

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 ... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

Obtain the corresponding Poisson-Boltzmann equation and the linearized version based on the Debye-Hiickel approximation. 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

When the electrolyte concentration is increased, the range of the double layer decreases dramatically (the Debye-Hiickel length decreases) and the magnitude of the surface potential also decreases. In the linear approximation, ]f(x) = iJj(c E)cxp( — (x-dB)/X) (for x>dB) and the second right-hand-side term of Eq. (48) becomes ... 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

This equation is precisely equal to that in the Debye-Hiickel treatment of ionic interaction for dilute electrolyte solutions14, only that the distance x refers to a central ion (point charge) and not to an electrode. In the Debye-Hiickel case, since the central ion is small and 0A small we can make the approximation (elinear approximation is not valid. 

With increase in salt concentration the approximations involved in the Debye-Hiickel theory become less acceptable. Indeed it is noteworthy that before this theory was published a quasi-lattice theory of salt solutions had been proposed and rejected (Ghosh, 1918). However, as the concentration of salt increases so log7 ,7 being the mean ionic activity coefficient, appears as a linear function of c1/3 (the requirement of a quasi-lattice theory) rather than c1/2, the DHLL prediction (Robinson and Stokes, 1959). Consequently, a quasi-lattice theory of salt solutions has attracted continuing interest (Lietzke et al., 1968 Desnoyers and Conway, 1964 Frank and Thompson, 1959 Bahe, 1972 Bennetto, 1973) and has recently received some experimental support (Neilson et al., 1975). 

The linearization of the PB equation is often called the Debye-Hiickel approximation and it is valid when qfo/kT < 1. At room temperature this corresponds to surface potentials, 0o, below 25 mV. In the case of flat surfaces and if symmetry is considered as in the Gouy-Chapman equation... 

The linearization that leads here to the Debye-Hiickel model is physically consistent in this argument. But the possibility of a model that is unlinearized in this sense is a popular query. More than one response has been offered including the (nonlinear) Poisson-Boltzmann theory and the EXP approximation see (Stell, 1977) also for representative numerical results for the systems discussed here. 

Equation (1.9) is the linearized Poisson-Boltzmann equation and k in Eq. (1.10) is the Debye-Htickel parameter. This linearization is called the Debye-Hiickel approximation and Eq. (1.9) is called the Debye-Hiickel equation. The reciprocal of k (i.e., 1/k), which is called the Debye length, corresponds to the thickness of the double layer. Note that nf in Eqs. (1.5) and (1.10) is given in units of m . If one uses the units of M (mol/L), then must be replaced by IQQQNAn, Na being Avogadro s number. 

Figure 1.4 shows y(x) for several values of yo calculated from Eq. (1.37) in comparison with the Debye-Hlickel linearized solution (Eq. (1.25)). It is seen that the Debye-Hiickel approximation is good for low potentials (lyol< 1). As seen from Eqs. (1.25) and (1.37), the potential i//(x) across the electrical double layer varies nearly... 

In this section, we present a novel linearization method for simplifying the nonlinear Poisson-Boltzmann equation to derive an accurate analytic expression for the interaction energy between two parallel similar plates in a symmetrical electrolyte solution [13, 14]. This method is different from the usual linearization method (i.e., the Debye-Hiickel linearization approximation) in that the Poisson-Boltzmann equation in this method is linearized with respect to the deviation of the electric potential from the surface potential so that this approximation is good for small particle separations, while in the usual method, linearization is made with respect to the potential itself so that this approximation is good for low potentials. 

In the usual Debye-Hiickel linearization approximation, Eq. (9.162) is linearized with respect to y itself, namely. 

The next-order correction terms to Derjaguin s formula and HHF formula can be derived as follows [13] Consider two spherical particles 1 and 2 in an electrolyte solution, having radii oi and 02 and surface potentials i/ oi and 1/ 02, respectively, at a closest distance, H, between their surfaces (Fig. 12.2). We assume that i/ oi and i//q2 are constant, independent of H, and are small enough to apply the linear Debye-Hiickel linearization approximation. The electrostatic interaction free energy (H) of two spheres at constant surface potential in the Debye-Hlickel approximation is given by... 

The simplest approach conventionally employed to describe the grain screening in colloidal plasmas is the Debye-Hiickel (DH) approximation, or, its modification for the case of the grain of finite size, the DLVO theory [6,7], The DH approximation represents the version of Poisson-Boltzmann (PB) approach linearized with respect to the effective potential based on the assumption that the system is in the state of thermodynamical equilibrium. The DH theory yields the effective interparticle interaction in the form of the so-called Yukawa potential which constitutes the basis for the Yukawa model. 

Table 3 Results of Debye-Hiickel linearization of Eq. (13) and Derjaguin approximation...

The third group of the approximate models includes various improvements of the Derjaguin approximation, linearization, and approximate solutions of PB Eq. (13) for spherical particles. The first improvement on the Derjaguin approximation for the interaction energy between identical spheres was probably obtained by the Debye-Hiickel linearization and the superposition approximation,given by ... 

Derjaguin approximation in the framework of the Debye-Hiickel linearization can be expressed... 

Here, g0 is the polyion s charge density, and the potential of the finite line charge in the Debye-Hiickel approximation is denoted by linear charge density and related to the Bjerrum length ZB via i s) = ZB. One finds that the solution to Eq. 1 for the linear charge density is [84,83] ... 

The Debye-Hiickel expression for p does not predict, even in its nonlinear form, a saturation effect for very large values oizexpjkTas would be desirable. Our equations show that for zetpjkT co, f and d are of the same order and thus in the limit pjzen+ —— 1. We encounter then the effect of saturation. In working with a linear approximation one cannot expect to find a saturation effect. 

The electrostatic interaction energy between diffused double layers based on the linear Debye-Hiickel approximation [Eq. (18)] for two parallel plates having constant surface charge are given by Usui in the form ... 

If the value of the electrostatic potential at the particle surface is low, V (I c) hQ/Rc C 1, the electrostatic potential at r > is even lower and the linearized form of the PB equation, often referred to as the Debye Hiickel (DH) approximation ... 

For further simplifications, one may note that for small values of the argument, it is possible to neglect higher-order terms in an exponential series and approximate exp(x) as 1 -1- x. This consideration, when appHed to the potential distributions depicted by Eqs. 12a and 12b, forms the basis of the Debye-Hiickel linearization principle [2], which effectively linearizes the pertinent exponential variation of ionic charge distribution for small values of e4>/k T. Under such approximations, Eq. 13 can be simplified as... 

The mathematical descriptions outlined so far for analyzing cases 1 and 2 implicitly assume the validity of the celebrated Debye-Hiickel linearization principle, as described earlier. However, for higher pH values (such as pH > 8), the surface potential may be such that the value of eij/lksT cannot be taken to be small at all locations. A limiting condition that constraints the applicability of the Debye-Hiickel approximation occurs for e j/lk T 1, which, for standard temperatures. 

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