Debye-Hiickel linearized solution

FIGURE 1.4 Potential distribution y x) = ze>J/ x)/kT around a positively charged plate with scaled surface potential yo = ze[j/JkT. Calculated foryo= U 2, and 4. Solid lines, exact solution (Eq. (1.37)) dashed lines, the Debye-Hiickel linearized solution (Eq. (1.25)). 

FIGURE 1.10 Scaled surface charge density scaled surface potential ya = ze>pJkT for a positively charged sphere in a symmetrical electrolyte solution of valence z for various values of ku. Solid line, exact solution (Eq. (1.86)) dashed line, Debye-Hiickel linearized solution (Eq. (1.76)). 

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... 

Figure 1.11 gives the scaled potential distribution y(r) around a positively charged spherical particle of radius a with yo = 2 in a symmetrical electrolyte solution of valence z for several values of xa. Solid lines are the exact solutions to Eq. (1.110) and dashed lines are the Debye-Hiickel linearized results (Eq. (1.72)). Note that Eq. (1.122) is in excellent agreement with the exact results. Figure 1.12 shows the plot of the equipotential lines around a sphere with jo = 2 at ka = 1 calculated from Eq. (1.121). Figures 1.13 and 1.14, respectively, are the density plots of counterions (anions) (n (r) = exp(+y(r))) and coions (cations) ( (r) = MCxp(—y(r))) around the sphere calculated from Eq. (1.121). 

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. 

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 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 ... 

With the Debye-Hiickel linearization the solution is simply... 

The potential near a protein in salt solution. Consider a protein sphere with a radius of 18 A, tind charge Q = -lOe, in an aqueous solution of 0.05 M NaCl at 25 °C. Consider the small ions as point charges and use the Debye-Hiickel linear approximation of the Poisson-Boltzmann equation. 

Another arena for the application of stochastic frictional approaches is the influence of ionic atmosphere relaxation on the rates of reactions in electrolyte solutions [19], To gain perspective on this, we first recall the early and often quoted triumph of TST for the prediction of salt effects, in connection with Debye-Hiickel theory, for reaction rates In kTST varies linearly with the square root of the solution ionic strength I, with a sign depending on whether the charge distribution of the transition state is stabilized or destabilized by the ionic atmosphere compared to the reactants. 

We know from experiment that log 7 is a linear function of I. The value of log 7 + is described well by the Debye-Hiickel limiting law in very dilute solution. Thus, we can substitute the expression... 

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. 

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 ... 

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... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

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. 

In a free solution, the electrophoretic mobility (i.e., peiec, the particle velocity per unit applied electric field) is a function of the net charge, the hydrodynamic drag on a molecule, and the properties of the solutions (viscosity present ions—their concentration and mobility). It can be expressed as the ratio of its electric charge Z (Z = q-e, with e the charge if an electron and q the valance) to its electrophoretic friction coefficient. Different predictive models have been demonstrated involving the size, flexibility, and permeability of the molecules or particles. Henry s theoretical model of pdcc for colloids (Henry, 1931) can be combined with the Debye-Hiickel theory predicting a linear relation between mobility and the charge Z ... 

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. 

If the solute is a salt, then the extrapolation to obtain, say, V3 can be based on the Debye-Hiickel limiting law (DHLL) or some variant of this equation. However, where non-polar solutes are concerned, there is no simple theory. It is generally assumed that the partial molar volume, V3 is a linear function of x3, and V3 is obtained by extrapolation to the value of V3 when x3 = 0 (Franks and Smith, 1968). 

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 electrostatic potential only can be determined relative to a reference point which normally is chosen to be zero at r — oo. However, this equation is still very difficult to solve and an analytical solutions are only available in special cases. Useful solutions occur at low surface potential, where the PB can be linearized (see Debye-Hiickel below). A famous analytical solution was derived by Gouy [12] and Chapman [13] independently (see below) for one flat surface in contact with an infinite salt reservoir. The interaction between two flat and charged surfaces in absence of salt, can also be solved analytically [14]. In other situations the nonlinearized PB equation has to be solved numerically. 

We might proceed by plotting versus m, drawing a smooth curve through the points, and constructing tangents to the curve at the desired concentrations in order to measure the slopes. However, for solutions of simple electrolytes, it has been found that many apparent molar quantities such as tp vary linearly with yfm, even up to moderate concentrations. This behavior is in agreement with the prediction of the Debye-Hiickel theory for dilute solutions. Since... 

An alternative approach is based on the view that the failure of the Debye-Hiickel theory at high concentrations stems from the fact that the development of the theory involved the linearization ofthe Boltzmann equation (see Section 3.3.5). If such a view is taken, there is an obvious solution to the problem instead of linearizing the... 

In most of the modem versions of the Debye-Hiickel theory of 1923, it is still assumed that the dielectric constant to be used is that of water. The dielectric constant of solutions decreases linearly with an increase in the concentration of the electrolyte. Using data in the chapter, calculate the mean activity coefficient for NaCl from 0.1 M to 2 M solutions, using the full equation with correction for the space taken up by the ions and the water removed by hydration. Compare the new calculation with those of Stokes and Robinson. Discuss the change in a you had to assume. 

Note that we can use a linear extrapolation because, at the low ionic strengths of these solutions, the Debye-Hiickel limiting law is quite sufficient. If all experimental data were of this quality, we could have pk -values listed to three significant decimal places ... 

---