Debye-Boltzmann equation

Under conditions of electrostatic interaction, the magnitude of the AGC° term varies with the salt concentration m, and hence, the dependence of the protein activity coefficient in the mobile phase can be expressed through the extended Debye-Boltzmann equation as... 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

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. 

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 

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... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

The theoretical inconsistencies inherent in the Poisson-Boltzmann equation were shown in Section 11.4 to vanish in the limit of very small potentials. It may also be shown that errors arising from this inconsistency will not be too serious under the conditions that prevail in many colloidal dispersions, even though the potential itself may no longer be small. Accordingly, we return to the Poisson-Boltzmann equation as it applies to a planar interface, Equation (29), to develop the Gouy-Chapman result without the limitations of the Debye-Hiickel approximation. 

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 ... 

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

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. 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

In the limit of an infinite micellar radius, i.e. a charged planar surface, the salt dependence of Ge is solely due to the entropy factor. A difficult question when applying Eq. (6.13) to the salt dependence of the CMC is if Debye-Hiickel correction factors should be included in the monomer activity. When Ge is obtained from a solution of the Poisson-Boltzmann equation in which the correlations between the mobile ions are neglected, it might be that the use of Debye-Hiickel activity factors give an unbalanced treatment. If the correlations between the mobile ions are not considered in the ionic atmosphere of the micelle they should not be included for the free ions in solution. 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

When Di > i>2, the effective Debye—Hiickel length X (which now depends on ip(x)) is larger than that obtained for the Poisson—Boltzmann equation. Consequently, the diffuse double layer is larger in the vicinity of a charged surface, as predicted earlier.4 7 9 However, when V2 > Vi (small counterions), X < X and the diffuse double layer is compressed. The effect is proportional to the ionic strength and is, in general, small for typical electrolyte concentrations, since n(v — v[Pg.337]

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... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

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... 

To make headway with the colloidal problem, the Poisson-Boltzmann equation must be solved in spherical coordinates. -> Debye and -> Huckel [iv] introduced the following approximation into the spherical case,... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

Statistical thermodynamics takes into account the microscopic structure of matter and how it is constituted. According to - Boltzmann the entropy can be correlated with the microscopic disorder of the system. Beside the Boltzmann distribution formula (- Debye-Htickel equation) other statistics (e.g., - Fermi-Dirac statistics) are of importance in electrochemistry (- semiconductors, -> BCS theory). 

Because this result has been obtained by solving a generalized Poisson-Boltzmann equation with the linearization approximation, it is necessary to compare it with the DLVO theory in the limit where the Debye approximation holds. In this case, Verwey and Overbeek [2], working in cgs (centimeter-gram-second) units, derived the following approximate equation for the repulsive potential ... 

Poisson-Boltzmann equation (ie., Debye-Huckel linearization) is... 

A Vital Step in the Debye-Huckel Theory of the Charge Distribution around Ions Linearization of the Boltzmann Equation... 

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... 

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. 

When the magnimde of the surface potential is arbitrary so that the Debye-Hiickel hnearization cannot be allowed, we have to solve the original nonlinear spherical Poisson-Boltzmann equation (1.68). This equation has not been solved but its approximate analytic solutions have been derived [5-8]. Consider a sphere of radius a with a... 

Here we treat a planar plate surface immersed in an electrolyte solution of relative permittivity e,. and Debye-Huckel parameter k. We take x- and y-axes parallel to the plate surface and the z-axis perpendicular to the plate surface with its origin at the plate surface so that the region z>0 corresponds to the solution phase (Fig. 2.1). First we assume that the surface charge density a varies in the x-direction so that a is a function of x, that is, cr = cr(x). The electric potential ij/ is thus a function of x and z. We assume that the potential i/ (x, z) satisfies the following two-dimensional linearized Poison-Boltzmann equation, namely,... 

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