Debye-Hiickel equation electrostatic potential

J.W. Krozel, D.A. Saville, Electrostatic interactions between two spheres solutions of the Debye-Hiickel equation with a charge regulation boundary condition. J. Colloid Interface Sci. 150(2), 365-373 (1992). doi 10.1016/0021-9797(92)90206-2 J. Lyklema, J.F.L. Duval, Hetero-interaction between Gouy-Stem double layers Charge and potential regulation. Adv. Colloid Interface Sci. 114—115, 27-45 (2005). doi 10.1016. cis. 2004.05.002... 

The electrostatic potential was assumed to obey the Debye-Hiickel equation... 

The presence of mutually repulsive electrostatic forces between bimo-lecukr suface membranes is established. However, weak van der Waals forces arising between neutral atoms in the bimolecular membrane are attractive. Of interest in this respect, is the distance relationship that exists between van der Waals and electrostatic forces between two adjacent membranes. It is known that the van der Waals forces will fall oflF as the inverse square of the distance, while the electrostatic force will vary exponentially with the distance between the membranes. The electrostatic double layer (a repulsive force) is of definite thickness and is dependent on ionic strength. At low surface potentials, this thickness is represented by the Debye-Hiickel equation ... 

The derivation of the Debye— Hiickel equation for the activity coefficient is based on the linearized Boltzmann equation for electrostatic charge distribution around an ion. This limits the applicability of Eq. (57) to solutes with low surface potentials, which occurs for solution concentrations of monovalent ions of < 0.01 M. However, it is important to note that die method used for deriving activity coefficient equation (25) is based on rigorous thermodynamics and is not limited by the Debye—Huckel theory. If, for example, the Gouy—Chapman equation [22] was... 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

The diffuse layer is described by the Gouy—Chapman theory of 1913 [21, 22], which is based on the same equations as the Debye—Hiickel theory of 1923 for electrolytes, which describes the electrostatic potential around an ion in a given ionic atmosphere [23]. 

Alternatively, there has been a revival of Debye-Hiickel (DH) theory [196-199] which provides an expression for the free energy of the RPM based on macroscopic electrostatics. Ions j are assumed to be distributed around a central ion i according to the Boltzmann factor exp(—/ , - y.(r)), where y(r) is the mean local electrostatic potential at ion j. By linearization of the resulting Poisson-Boltzmann (PB) equation, one finds the Coulombic interaction to be screened by the well-known DH screening factor exp(—r0r). The ion-ion contribution to the excess free energy then reads... 

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. 

The Debye-Hiickel theory focuses on and by using Poisson s equation of electrostatics finds an explicit equation for yfrj, firom which the potential, at the surface of the central j ion... 

Since the first measurements of the electrostatic double-layer force with the AFM not even 10 years ago, the instrument has become a versatile tool to measure surface forces in aqueous electrolyte. Force measurements with the AFM confirmed that with continuum theory based on the Poisson-Boltzmann equation and appKed by Debye, Hiickel, Gouy, and Chapman, the electrostatic double layer can be adequately described for distances larger than 1 to 5 nm. It is valid for all materials investigated so far without exception. It also holds for deformable interfaces such as the air-water interface and the interface between two immiscible liquids. Even the behavior at high surface potentials seems to be described by continuum theory, although some questions still have to be clarified. For close distances, often the hydration force between hydrophilic surfaces influences the interaction. Between hydrophobic surfaces with contact angles above 80°, often the hydrophobic attraction dominates the total force. 

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 ... 

Coimterion condensation has detractors (28-34), who point to flaws in the concept s derivation, such as artificial subdivision of the counterions into two populations, inappropriate extrapolation of the Debye-Hiickel approximation to regions of high electrostatic potential, and inconsistent treatment of counterions. The full nonlinear Poisson-Boltzmann equation offers a more rigorous way to interpret electrostatic phenomena in electrolyte solutions, but the physical picture obtained through this equation is different in some ways from the one suggested by condensation (21,34,35). In particular, a Poisson-Boltzmann analysis does not readily identify distinct populations of condensed and free counterions but rather a smoothly varying Gouy-Chapman layer. Nevertheless, Poisson-Boltzmann-based... 

The concept behind the DH theory was not new, in that Milner (3a), almost a decade before, formulated a theory of ionic solutions based on the concept of "ionic atmosphere". He, however, was unable to solve the proposed equations. Double layer theories(3b,3c), which used the same concept, also preceded the DH theory. The merit of Debye and Hiickel was to introduce several approximations that made an analytical solution for the theory possible. The starting point of the DH theory is the assumption that the excess of thermodynamic properties of electrolyte solutions (when compared with non-electrolyte solutions) is due only to the Coulombic interactions between the ions. It is then necessary to calculate the average electrostatic potential at the surface of a given ion (taken as reference) due to all the other ions. These other ions constitute the "ionic atmosphere". Once this potential is known, it is evidently possible to calculate all the thermodynamic properties of the system. Indicating with z e and zje the charge of the reference ion (i) and of an arbitrary ion (j) in the "ionic atmosphere", respectively, the effective interaction energy between the two ions will be... 

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-Hiickel theory of electrolyte solutions. It is derived from the classical Poisson equation for the electrostatic potential... 

The electrostatic potential energy between two ions given by Equations 3.19 and 3.20 is called the Debye-Hiickel potential energy. The collective effect of the ions in the solution is to screen the Coulomb interaction between a pair of ions given by Equation 3.5 resulting in the screened electrostatic interaction given by Equation 3.19. 

The ion cloud in the solution modifies the result of Equation 3.77. Exact solution of the Poisson-Boltzmann equation (Equation 3.70) is known for the salt-free solutions containing only the counterions (Alfrey et al. 1951). As expected, the electric potential falls off smoothly with the radial distance, and there exists a counterion cloud near the cylinder. In order to get insight into the basic nature of the electrostatics in salty electrolyte solutions around a charged thin cylinder, we linearize Equation 3.70 to get the Debye-Hiickel theory (Equation 3.71). Solving this equation with the boundary conditions that the electric field vanishes far away from the cylinder and that it is given by Equation 3.76 at the surface of the cylinder, the result is... 

We will not discuss the details of the Debye-Huckel theory. The main idea of the theory was to pretend that the ions in a solution could have their charges varied reversibly from zero to their actual values. This charging process created an ion atmosphere around a given ion with an excess of ions of the opposite charge. The reversible net work of creating the ion atmosphere was calculated from electrostatic theory. According to Eq. (4.1-32) the reversible net work is equal to AG, which leads to equations for the electrostatic contribution to the chemical potential and the activity coefficient for the central ion. The principal result of the Debye-Hiickel theory is a formula for the activity coefficient of ions of type i ... 

Equation (62) is rigorous (assuming Gibbs and Helmholtz free energies are the same for ions in a solid—solution interfacial system) but requires explicit electrostatic potential equations in order to obtain an explicit activity coefficient equation. Since rigorous, explicit electrostatic potential equations for solute and surface site ions have yet to be derived, the approximate electrostatic potential equations, which are solutions to the linearized Poisson—Boltzmann equation, were used here and by Debye—Hiickel to give... 

Equation (104) is a rigorous mass action equation in terms of explicit functions except for the electrostatic potentials. Electrostatic potentials are, at best, difficult to measure and no rigorous explicit equation expressing the electrostatic potential of a solute ion has yet been developed. Instead, the approximate explicit equation (60) (based on Debye-Hiickel theory) for the electrostatic potential of a solute ion will be used to write equations for each of die activity coefficients in Eq. (104) ... 

The replacement of the potential of mean force with the mean electrostatic potential by Debye and Hiickel (and implicit in the Gouy-Chapman approach) has caused the greatest amount of concern for those applying the PB equation. Fowler severely criticized use of the PB equation on this basis, but his investigation was soon shown to be overly restrictive.Still, the effect of neglecting ion-ion correlation, which this mean-field approximation implies, is a continual source of study. Hence there have been published numerous comparisons between PB theory and more detailed statistical-mechanical theories or calculations that do include correlation. While the size of the effect depends on the particular system studied, calculations on the cylindrical and all-atom models of DNA show that PB calculations tend to underestimate ion concentrations at the surface by 15-25% for mono- or divalent ions, respec-tively. " "- ... 

In addition to neglecting ion correlation, using the mean electrostatic potential has the undesirable consequence that the (nonlinear) PB equation no longer satisfies a reciprocity condition that use of the potential of mean force would obey. Linearization of the equation by Debye and Hiickel regained this condition. These considerations led Outhwaite and others to propose modifications of the PB equation to treat these problems. Within this modified Poisson-Boltzmaim (MPB) theory, the effect of ion correlation is expressed in terms of a fluctuation potential for which a first-order (local) expression, written as an activity coefficient, can be derived. Their result for bulk hard-sphere electrolyte ions of valence z, and common radius a gives the formula ... 

The real behavior of systems is described by the activity coefficient y,. Instead of the concentration C of a dissolved species, one uses the activity a, = c y,. In the light of the Debye-Hiickel theory, y takes care of the electrostatic interactions of the ions. This is the main interaction for charged species in comparison with the smaller dipole and Van der Waals forces, which may be important in the case of uncharged species, but which are not included in the Debye-Huckel theory. The chemical potential p depends on the concentration according to Equation 1.37. 

Fig. 4. Comparison between the electrostatic potentials around a rod-like polyion calculated with and without the Debye-Hiickel approximation. The broken lines denote the calculated values of Equation (7), while the solid lines denote the values calculated from the Poisson-Boltzmann equation without the Debye-Huckel approximation. (Reproduced from Reference [6].)...

The electrostatic double-layer force can be calculated using the continuum theory, which is based on the theory of Gouy, Chapman, Debye, and Hiickel for an electrical double layer. The Debye length relates the surface charge density of a surface to the electrostatic surface potential /o via the Grahame equation, which for 1 1 electrolytes can be expressed as... 

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