Double layer Stem plane

The evolution of the potential, when we move away from the particle surface, is calculated by expressing the electrochemical potential difference between the particle surface and the solution. In the structured part of the double layer (Stem layer), the ions are supposed to be located on distinct planes. The decreases in the potential between the surface and the IHP, and between the IHP and the OHP are therefore as linear as inside a condenser. The potential decreases exponentially with the distance in the diffuse layer starting from the OHP. The electrically disturbed zone extends to about 100 angstroms. Its thickness depends highly on the concentration and the charge of the electrolyte ions. The greater the charge and the concentration, the more the diffuse layer is compressed. 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

The electroosmotic pumping is executed when an electric field is applied across the channel. The moving force comes from the ion moves in the double layer at the wall towards the electrode of opposite polarity, which creates motion of the fluid near the walls and transfer of the bulk fluid in convection motion via viscous forces. The potential at the shear plane between the fixed Stem layer and Gouy-Champmon layer is called zeta potential, which is strongly dependent on the chemistry of the two phase system, i.e. the chemical composition of both solution and wall surface. The electroosmotic mobility, xeo, can be defined as follow,... 

For present purposes, the electrical double-layer is represented in terms of Stem s model (Figure 5.8) wherein the double-layer is divided into two parts separated by a plane (Stem plane) located at a distance of about one hydrated-ion radius from the surface. The potential changes from xj/o (surface) to x/s8 (Stem potential) in the Stem layer and decays to zero in the diffuse double-layer quantitative treatment of the diffuse double-layer follows the Gouy-Chapman theory(16,17 ... 

For a long time, the electric double layer was compared to a capacitor with two plates, one of which was the charged metal and the other, the ions in the solution. In the absence of specific adsorption, the two plates were viewed as separated only by a layer of solvent. This model was later modified by Stem, who took into account the existence of the diffuse layer. He combined both concepts, postulating that the double layer consists of a rigid part called the inner—or Helmholtz—layer, and a diffuse layer of ions extending from the outer Helmholtz plane into the bulk of the solution. Accordingly, the potential drop between the metal and the bulk consists of two parts ... 

The variation of the electric potential in the electric double layer with the distance from the charged surface is depicted in Figure 6.2. The potential at the surface ( /o) linearly decreases in the Stem layer to the value of the zeta potential (0- This is the electric potential at the plane of shear between the Stern layer (and that part of the double layer occupied by the molecules of solvent associated with the adsorbed ions) and the diffuse part of the double layer. The zeta potential decays exponentially from to zero with the distance from the plane of shear between the Stern layer and the diffuse part of the double layer. The location of the plane of shear a small distance further out from the surface than the Stem plane renders the zeta potential marginally smaller in magnitude than the potential at the Stem plane ( /5). However, in order to simplify the mathematical models describing the electric double layer, it is customary to assume the identity of (ti/j) and The bulk experimental evidence indicates that errors introduced through this approximation are usually small. 

Fig. 1.10 Schematic view of the electrical double layer in agreement with the Gouy-Chapman-Stem-Grahame models. The metallic electrode has a negative net charge and the solvated cations define the inner limit of the diffuse later at the Helmholtz outer plane (OHP). There are anions adsorbed at the electrode which are located at the inner Helmholtz plane (IHP). The presence of such anions is stabilized by the corresponding images at the electrode in such a way that each adsorbed ion establishes the presence of a surface dipole at the interface...

The inner part of the double layer may include specifically adsorbed ions. In this case, the center of the specifically adsorbed ions is located between the surface and the Stem plane. Specifically adsorbed ions (e.g., surfactants) either lower or elevate the Stem potential and the zeta potential as shown in Figure 4.31. When the specific adsorption of the surface-active or polyvalent counter ions is strong, the charge sign of the Stem potential will be reversed. The Stem potential can be greater than the surface potential if the surface-active co-ions are adsorbed. The adsorption of nonionic surfactants causes the surface of shear to be moved to a much longer distance from the Stem plane. As a result, the zeta potential will be much lower than the Stem potential. 

The potential in the diffuse layer decreases exponentially with the distance to zero (from the Stem plane). The potential changes are affected by the characteristics of the diffuse layer and particularly by the type and number of ions in the bulk solution. In many systems, the electrical double layer originates from the adsorption of potential-determining ions such as surface-active ions. The addition of an inert electrolyte decreases the thickness of the electrical double layer (i.e., compressing the double layer) and thus the potential decays to zero in a short distance. As the surface potential remains constant upon addition of an inert electrolyte, the zeta potential decreases. When two similarly charged particles approach each other, the two particles are repelled due to their electrostatic interactions. The increase in the electrolyte concentration in a bulk solution helps to lower this repulsive interaction. This principle is widely used to destabilize many colloidal systems. 

Earlier theories by Gouy, Chapman, and Hcrzfeld discussed the double layer as wholly of this diffuse type but Stem points out that these give far too high values for the capacity of the double layer, partly because in them the ions are supposed mathematically to be able to approach indefinitely close to the solid surface, which is impossible physically owing to the size of the ions. Stern s theory gives a complicated expression for the capacity of the double layer, but accounts reasonably well for the experimental values. Though the layer is largely diffuse in many cases, the capacity is usually of the same order as if the layer were of the plane parallel type, because most of the ions are fairly close to the fixed part of the layer. 

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

FIGURE 9,14 Schematic representation of molecular arrangement dose to a solid surface showing the inner (IHP) and outer (OHP) Helmholtz planes, the stem layer, difAise double layer, also called the Gouy layer, and the slip plane where the zeta potential is measured. Also shown is the potential for various distances from the surface. 

The adsorbed Stem layer is compensated by a compact and essentially fixed layer of hydrated counterions and water molecules which takes the form of a molecular capacitor between the inner and outer Helmholtz planes shown in Figure 9.14. The solid surface adsorbs the Stem layer ions and gives a potential of the inner Helmholtz plane, which is partially compensated by the hydrated counterions and water molecviles of the outer Helmholtz plane. The diffuse double layer of (jOuy-Chapman starts at the OHP and extends further into the liquid. 

Stem layers can be introduced In categories of proficiency of which three are drawn in fig. 3.20a, b and c, and one in fig. 3.21. Figure 3.20a is the most simple picture only ion size is accounted for. and only in the first layer. We shall refer to this picture as the zeroth-order Stern layer. Even in this simple case the double layer is actually a triple layer. The charge distribution remains ideal, meaning that all the relevant equations of sec. 3.5 remain valid after replacing x by (x-d). The borderline between the Stem layer of thickness d and the diffuse layer is called the outer Helmholtz plane (oHp). The charge balance is simply... 

The Gouy-Chapman theory provides a better approximation of reality than does the Helmholtz theory, but it still has limited quantitative application. It assumes that ions behave as point charges, which they cannot, and it assumes that there is no physical limit for the ions in their approach to the TPB, which is not true. Stem, therefore, modified the Gouy-Chapman diffuse double layer. His theory states that ions do have finite size, so they cannot approach the TPB closer than a few nm [54, 60], The first ions of the Gouy-Chapman diffuse double layer are in the gas phase but not at the TPB. They are at some distance 8 away from the zirconia-metal-gas interface. This distance will usually be taken as the radius of the ion. As a result, the potential and concentration of the diffuse part of the layer are low enough to justify treating the ions as point charges. Stem also assumed that it is possible that some of the ions are specifically adsorbed by the TPB in the plane 8, and this layer has become known as the Stem layer. Therefore, the potential will drop by T o - Pg over the molecular condenser (i.e., the Helmholtz plane) and by T g over the diffuse layer. Pg has become known as the zeta (Q potential. 

Both the inner and outer Helmholtz plane are the Stem layer or Stem double layer. The diffuse layer consists of... 

In the Stem-Gouy-Chapman (SGC) theory the double layer is divided into a Stem layer, adjacent to the surface with a thickness d, and a diffuse layer of point charges. The diffuse layer begins at the Stem plane in a distance d, from the surface. In the simplest case the Stem layer is free of charges. In real cases the Stem layer is formed by specifically adsorbed ions. The condition of electroneutrality was given by Eq. (2.59) In addition to and Oj, the surface charge can be represented by the Stem potential. It transforms the conditions of electroneutrality into the equation for the determination of the Stem potential. 

A distinction is often made between the plane where the centres of charge of the partially dehydrated specifically adsorbed ions reside, the inner Helmholtz plane, and Stem plane at distance d, which is also called the outer Helmholtz plane. The double layer model consists of an inner and outer Helmholtz layer and a diffuse layer. This is often called the triple layer model. 

Figure 3. Highly schematic view of the electrical double layer (EDL) at a metal oxide/aqueous solution interface showing (1) hydrated cations specifically adsorbed as inner-sphere complexes on the negatively charged mineral surface (pH > pHpzc of the metal oxide) (2) hydrated anions specifically and non-specifically adsorbed as outer-sphere complexes (3) the various planes associated with the Gouy-Chapman-Grahame-Stem model of the EDL and (4) the variation in water structure and dielectric constant (s) of water as a function of distance from the interface, (from Brown and Parks 2001, with permission)...

A schematic representation of the inner region of the double layer model is shown in Fig. 1. Figure lb describes the distribution of counterions and the potential profile /(a ) from a positively charged surface. The potential decay is caused by the presence of counterions in the solution side (mobile phase) of the double layer. The inner Helmholtz plane (IHP) or the inner Stem plane (ISP) is the plane through the centers of ions that are chemically adsorbed (if any) on the solid surface. The outer Helmholtz plane (OHP) or the outer Stem plane (OSP) is the plane of closest approach of hydrated ions (which do not adsorb chemically) in the diffuse layer. Therefore, the plane that corresponds to x = 0 in Eq. (4) coincides with the OHP in the GCSG model. The doublelayer charge and potential are defined in such a way that ao and /o, op and Tp, and <5d and /rf are the charge densities and mean potentials of the surface plane, the Stem layer (IHP), and the diffuse layer, respectively (Fig. 1). 

Three interface layers occur within the electrical or the diffuse double layer (DDL) of a clay particle the inner Helmholtz plane (IHP) the outer Helmholtz plane (OHP) with constant thicknesses of Xi and X2, respectively and third is the plane of shear where the electro kinetic potential is measured (Rg. 2.10). This plane of shear is sometimes assumed to coincide with the OHP plane. The IHP is the outer limit of the specifically adsorbed water, molecules with dipoles, and other species (anions or cations) on the clay solid surface. The OHP is the plane that defines the outer limit of the Stem layer, the layer of positively charged ions that are condensed on the clay particle surface. In this model, known as the Gouy-Chapman-Stera-Grahame (GCSG) model, the diffuse part of the double layer starts at the location of the shear plane or the OHP plane (Hunter, 1981). The electric potential drop is linear across the Stem layer that encompasses the three planes (IHP, OHP, and shear planes) and it is exponential from the shear plane to the bulk solution, designated as the reference zero potential. 

Fig. 5.15 Electric double layer surrounding the particle in the solution 1 rigid Stem s layer, 2 diffusion layer, 3 shear plane...

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