Poisson-Boltzmann equation, interface between

Derive and solve the appropriate linear Poisson-Boltzmann equation for the interface between two immiscible solutions. 

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. 

Gouy and Chapman (1910-13) independently used the Poisson-Boltzmann equations to describe the diffnse electrical double-layer formed at the interface between a charged snrface and an aqueous solution. 

In the paragraphs below, we first examine the simple, analytical results that can be derived from the linear Poission-Boltzmann equation for a single particle interacting with a flat surface. Next, more complicated physical situations are considered, including interactions between many particles and a wall between a particle and a deformable interface between a protein and a wall and between a moving particle and a wall. In Sec. Ill, solutions to the nonlinear Poisson-Boltzmann equation are considered, and comparisons are made between the linear and nonlinear versions and also with more... 

A hierarchy of approximations now exists for calculating interactions between a charged particle and a charged, planar interface in electrolyte solutions. At moderate surface potentials less than approximately 2(kT/e the linear Poisson-Boltzmann equation provides a good approximation in many circumstances, provided the solution is a 1 1 electrolyte at low to moderate ionic strength. The relative simplicity of the linear equation makes it particularly useful for examining problems that are complicated in other ways, such as interactions involving many particles, interactions with deformable interfaces, and interactions where the detailed structure and properties of the particle (or macromolecule) play an important role. 

A thorough discussion of the basic theory describing electrostatic interactions can be found in [7] the pertinent points are discussed below. Electrostatic forces arise from the osmotic pressure difference between two charged surfaces as a result of the local increase in the ionic distribution around each charged surface. For a single electrified interface, the local ion distribution is coupled to the potential distribution near that surface and can be described using the Poisson-Boltzmann equation. The solution of this equation shows that for low surface potentials the potential follows an exponential function with distance from the interface, D, given by... 

The electrostatic interaction between two flat surfaces is an extension of the single electrified interface where the ion distributions overlap at close separations. The two-surface case can also be described according to the Poisson-Boltzmann equation where the ion distributions for each charged interface are coupled. A complete solution to... 

Surface potentials at the electrode-solution interface have been described by a number of formalisms. The most successful of these was offered originally by Gouy and Chapman with subsequent elaborations from Chapman, Stem, Bockris etc. (outlined in ref 1 21). McLaughlin (22) and others (outlined in 1) suggested that a combination of the Poisson and Boltzmann equations best describes the state of affairs in the space between the membrane surface and the bulk phase aqueous solution ie. the electrode-water interface. The Poisson-Boltzmann equation, with defined boundary conditions can be solved analytically (1,22) to yield an expression for the surface potential as follows ... 

Dependencies of AG on the average separation l between SC>3 groups and on the distance a between the plane of proton transfer and the negatively charged interface were studied theoretically in Ref. 43, 44. Using an approach based on the Poisson-Boltzmann equation, modulations of the electrostatic potential and of the distribution of mobile protons in the proximity of the charged anionic sites were calculated. 

Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

Chan, B.K.C. Chan, D.Y.C. Electrical doublelayer interaction between spherical colloidal particles an exact solution. J. Colloid Interface Sci. 1983, 92 (1), 281-283 Palkar, S.A. Lenhoff, A.M. Energetic and entropic contributions to the interaction of unequal spherical double layers. J. Colloid Interface Sci. 1994, 165 (1), 177-194 Qian, Y. Bowen, W.R. Accuracy assessment of numerical solutions of the nonlinear Poisson-Boltzmann equation for charged colloidal particles. J. Colloid Interface Sci. 1998, 201 (1), 7-12 Carnie, S.L. Chan, D.Y.C. Stankovich, J. Computation of forces between spherical colloidal particles Nonlinear Poisson-Boltzmann theory. J. Colloid Interface Sci. 1994, 165 (1), 116-128 Stankovich, J. Carnie, S.L. Electrical double layer interaction between dissimilar spherical colloidal particles and between a sphere and a plate nonlinear Poisson-Boltzmann theory. Langmuir 1996,12 (6), 1453-61. 

The concept of capillary waves can be used to explain how the surface roughness increases the interfacial capacity beyond the Verwey-Niessen value. For this purpose, Pedna and Badiali [82] have solved the linear Poisson-Boltzmann equation across the interface between two solutions vyith different dielectric constants and Debye lengths separated by a corrugated surface. A major difficulty is the boundary condition at the rough interface. 

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. 

Grahame equation and also as the contact theorem [6]. This, fundamentally, is a relationship between the surface charge density, (Tq (which is defined as o-q = — Jpedy, with a SI unit of C/m ), and the limiting value of the ionic density profile at the substrate-fluid interface. For a single fiat surface with an infinite extent of the adjacent liquid, an expression for co can be obtained from the Poisson-Boltzmann equation as... 

In 2006, the first x-ray reflectivity study of an ITIES was published in a series of papers by Luo et al. [68-70]. They studied an interface between a nitrobenzene solution of tetrabutylammonium tetraphenylborate (TBATPB) and an aqueous solution of tetrabutylammonium bromide (TBABr). The concentration of TBABr was varied to control the Galvani potential difference using an experimental setup as shown in Eigure 1.4. The ion distributions were predicted by a Poisson-Boltzmann equation, that explicitly includes a free energy profile for ion transfer across the interface described either by a simple analytic form or by a potential of mean force from molecular dynamics simulations. 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

A. B. Glendinning and W. B. Russel,/. Colloid Interface Sci., 93,95 (1983). The Electrostatic Repulsion between Charged Spheres from Exact Solutions to the Linearized Poisson-Boltzmann Equation. 

Ions in Clays, Fig. 2 Electro-osmotic flow profile between two Na-montmorillonite surfaces separated by a 4.5-mn pore containing water. Reference molecular dynamics (MD) simulations allow to test the validity of continuous models based on the Navier-Stokes (NS) and Poisson-Boltzmann (PB) equations. Such equations must be solved for given boundary conditions (stick or slip) at the solid/liquid interface... 

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