Charged planar surfaces

FIG. 1 Geometries of electrolyte interfaces, (a) A planar electrode immersed in a solution with ions, and with the ion distrihution in the double layer, (b) Particles with permanent charges or adsorbed surface charges, (c) A porous electrode or membrane with internal structures, (d) A polyelectrolyte with flexible and dynamic structure in solution, (e) Organized amphophilic molecules, e.g., Langmuir-Blodgett film and microemulsion, (f) Organized polyelectrolytes with internal structures, e.g., membranes and vesicles. 

A question of practical interest is the amount of electrolyte adsorbed into nanostructures and how this depends on various surface and solution parameters. The equilibrium concentration of ions inside porous structures will affect the applications, such as ion exchange resins and membranes, containment of nuclear wastes [67], and battery materials [68]. Experimental studies of electrosorption studies on a single planar electrode were reported [69]. Studies on porous structures are difficult, since most structures are ill defined with a wide distribution of pore sizes and surface charges. Only rough estimates of the average number of fixed charges and pore sizes were reported [70-73]. Molecular simulations of nonelectrolyte adsorption into nanopores were widely reported [58]. The confinement effect can lead to abnormalities of lowered critical points and compressed two-phase envelope [74]. 

FIG. 13 Contributions to the pressure between two —0.244 C/m charged planar surfaces separated by a 0.1 molar 2 2 RPM electrolyte. The open squares, circles, down-triangles, and diamonds are the kinetic, collision, electrostatic, and total pressures, respectively, from results of VaUeau et al. [98]. The corresponding solid symbols are unpublished results of Lee and Chan. The lines are calculations by the hypematted-chain (HNC) equation. 

We shall use the familiar Gouy-Chapman model (3 ) to describe the behaviour of the diffuse double lpyer. According to this model the application of a potential iji at a planar solid/electrolyte interface will cause an accumulation of counter-ions and a depletion of co-ions in the electrolyte near the interface. The disposition of diffuse double layer implies that if the surface potential of the planar interface at a 1 1 electrolyte is t ) then its surface charge density will be given by ( 3)... 

By preparing planar lipid monolayers or bilayers on hydrophobically derivatized or native hydrophilic glass, respectively, the adsorption equilibrium constants of a blood coagulation cascade protein, prothrombin, have been examined by TIRF on a surface that more closely models actual cell surfaces and is amenable to alterations of surface charge. It was found that membranes of phosphatidylcholine (PC) that contain some phosphatidyl-serine (PS) bind prothrombin more strongly than pure PC membranes/83... 

STM has also been used to examine the dynamics of potential-dependent ordering of adsorbed molecules [475-478]. For example, the reversible, charge-induced order-disorder transition of a 2-2 bipyridine mono-layer on Au(lll) has been studied [477]. At positive charges, the planar molecule stands vertically on the surface forming polymeric chains. The chains are randomly oriented at low surface charge but at higher potentials organize into a parallel array of chains, which follow the threefold symmetry of the Au(l 11) substrate as shown in Fig. 34. Similar results were found for uracil adsorption on Au(lll) and Au(lOO) [475,476]. 

The Gouy-Chapman model assumes (1) the exchangeable cations exist as point charges, (2) colloid surfaces are planar and infinite in extent, and (3) surface charge is distributed uniformly over the entire colloid surface. Even though this assumption... 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

FIGURE 15.10 Plots of the Gibbs free energy per unit area, AG/A, as a function of the distance between two oppositely charged planar surfaces, L, with the ionic strength as a parameter. The curves are calculated from Equation 15.63 with e=80, c =-0.16C/m, and Op = 0.03C/m. ... 

Retention of proteins in ion exchange chromatography is mainly caused by electrostatic effects. Because both the protein and the surface have an electrical double layer associated to it, there is an increase in entropy when the two surfaces approach each other. This is due to a release of counter ions from the two double layers when they overlap. The model that is discussed here is based on a solution of the linearized Poisson-Boltzmann for two oppositely charged planar surfaces. We also show the result from a model where the protein is considered as a sphere and the... 

Hence, the surface charge can be easily calculated from the surface potential for a planar surface. 

Additional experiments were performed in 2001 in the same laboratory by Storm et al. [59], in which longer DNA molecules (>40 nm) with various lengths and sequence compositions were stretched on the surface between planar electrodes in various configurations (see Fig. 5). No current was observed in these experiments suggesting that charge transport through DNA molecules longer than 40 nm on surfaces is blocked. 

The deficiency of positive ions (T+ < 0) adjacent to a positively charged planar surface may be evaluated as follows (y = ze //kBT) ... 

The aim of this chapter is to calculate the electric potential near a charged planar interface. In general, this potential depends on the distance normal to the surface x. Therefore, we consider a planar solid surface with a homogeneously distributed electric surface charge density a, which is in contact with a liquid. The surface charge generates a surface potential... 

This simple equation is, however, only valid for R Xp- If the radius is not much larger than the Debye length we can no longer treat the particle surface as an almost planar surface. In fact, we can no longer use the Gouy-Chapman theory but have to apply the theory of Debye and Hiickel. Debye and Hiickel explicitly considered the electric double layer of a sphere. A result of their theory is that the total surface charge and surface potential are related by... 

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. 

Fig. 6.3. The calculated reduced surface potential = e(ri)/kT versus the logarithm of the amphiphile concentration C (M) with no salt added for a spherical, cylindrical and planar aggregate. The surface charge density has been chosen as fixed at a8 = 0.228 Cm- 2. The radii of the sphere and the cylinder are 1.8 nm...

The photoinduction ion flux derives from the similarity of vesicle systems to the proton flux in halobacterium halobium cell envelopes in the bacteriorhodopsin photocycle [126]. Liposome permeability to glucose can similarly be induced by photoexdtation in vesicles containing polyacetylene or thiophene as ion mediators [127]. As in planar bilayers, the surface charge [128] of the vesicle and the chain length of the component surfactant [129] influence assodation between the donor-acceptor pairs, and hence the distance of separation of components inside and outside the vesicle walls. 

Artificial asymmetric membranes composed of outer membranes of various species of Gram-negative bacteria and an inner leaflet of various phospholipids have been prepared using the Montal-Mueller technique [65]. Such planar bilayers have been used, for example, to study the molecular mechanism of polymyxin B-mem-brane interactions. A direct correlation between surface charge density and self-promoted transport has been found [66]. 

The condensed fraction, of density nc, is localized on a two-dimensional (2D) planar surface [51]. The fraction of condensed counterions in 2D is x = Z0enc/ao, and the reduced surface charge density is enR = u0 — Z0enc. The free energy of the condensed counterions per unit area is [51]... 

In this Appendix, equations will be derived for the double layer interaction between two charged, planar surfaces in an electrolyte-free system. W e assume that the potential ip(x) obeys the Poisson—Boltzmann equation... 

Whereas the charging approach could be applied to any geometry but only at constant surface charge or potential, the Langmuir expression could be employed for any surface conditions but only for parallel planar plates. The addition of electrostatic, entropic, and chemical contributions would allow the calculation of the free energy of interaction for systems of any shape and any surface conditions, if one could derive a general expression for the chemical free energy contribution. 

Figure 2. The interaction free energy (per unit area) as a function of the separation distance l for two identical plates, planar and parallel, at T = 300 K, < = 80, ce = 0.01 M and (a) Kb = 1.0 M, N = 6.25 x 1015 sites/m2 (b) Kb =1.0 M, N = 6.25 x 1016 sites/m2 (c)Z2d = 0.001 M,ZV= 3.125 x 1017 sites/m2 (d)Z2b = 0.001 M,N= 3.125 x 1018 sites/m2. The continuous thick line represents the exact result the up triangles represent the upper bound, the down triangles represent the lower bound, the circles represent the constant surface charge approximation , and the crosses represent the constant surface potential approximation .

The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions). It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used. 

Another difficulty in using the Deijaguin approximation arises when the charge is related to the surface potential via various ionic equilibria. Indeed, in the linear approximation, the surface potential is related to the surface charge density, o, of a single planar surface by... 

For the moment let us follow closely the analysis of Richmond [68,69] who considered two planar surfaces, one at z = 0 and one at z = h, with arbitrary nonuniform distributions of charge or potential y s) and y2(s). The boundaries mark the ends of two infinite planar half spaces with dielectric permittivities, g] and S3, separated by a third, intermediate dielectric continuum of width h and permittivity, e2. This central medium contains a simple electrolyte solution. The generic function, y(s), we use to represent either a surface potential, P(s), or a surface charge, cr(s). s = (x,.y) is again the position vector in the plane of a surface. An arbitrary source distribution can be represented by the Fourier integral,... 

FIG. 4 Comparison between MGC theory (solid curves) and Monte Carlo simulation (circles and triangles) of the diffuse-ion swarm on a planar charged surface. Distributions of cations (c+) and anions (c ) are shown for a 1 1 electrolyte solution and two surface charge densities (oq). 

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