Double layer shear plane

The electrokinetic potential may thus be viewed as the potential at some plane, referred to as the plane of shear (the slipping plane), located within the limits of the diffuse part of the double layer. The plane of shear separates the immobilized part of the liquid phase bound to the solid surface from the remaining mobile part in which the displacement takes place. The curve describing the change in the displacement velocity of the layers of liquid as a function of the distance from the wall, u(x), matches the x axis up to the plane of shear, and at x>A has the same shape as the function showing the change in the potential as a function of distance (see Figs. V-7 and V-8). 

In the case of a charged particle, the total charge is not known, but if the diffuse double layer up to the plane of shear may be regarded as the equivalent of a parallel-plate condenser, one may write... 

Fig. 8. Electrical double layer of a sohd particle and placement of the plane of shear and 2eta potential. = Wall potential, = Stern potential (potential at the plane formed by joining the centers of ions of closest approach to the sohd wall), ] = zeta potential (potential at the shearing surface or plane when the particle and surrounding Hquid move against one another). The particle and surrounding ionic medium satisfy the principle of electroneutrafity.

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

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,... 

The charge or zeta ( ) potential of the filler particle (i.e. the charge at the plane of shear between the particle s diffuse double layer and the bulk liquid phase) can be obtained by measuring its mobility in an applied electric field of known magnitude. The mobility is a function of the field gradient and is therefore expressed as a speed per unit potential gradient (/im/s/V/cm). Mobility and therefore zeta potential are both a function of pH (Figure 6.4). 

The electrokinetic potential (zeta potential, Q is the potential drop across the mobile part of the double layer (Fig. 3.2c) that is responsible for electrokinetic phenomena, for example, elecrophoresis (= motion of colloidal particles in an electric field). It is assumed that the liquid adhering to the solid (particle) surface and the mobile liquid are separated by a shear plane (slipping plane). The electrokinetic charge is the charge on the shear plane. 

The surface potential is not accessible by direct experimental measurement it can be calculated from the experimentally determined surface charge (Eqs. 3.1 - 3.3) by Eqs. (3.3a) and (3.3b). The zeta potential, calculated from electrophoretic measurements is typically lower than the surface potential, y, calculated from diffuse double layer theory. The zeta potential reflects the potential difference between the plane of shear and the bulk phase. The distance between the surface and the shear plane cannot be defined rigorously. 

Effect of adsorbed polymer on the double-layer. Because of the presence of adsorbed train segments, the double layer is modified. The zeta-potential, , is displaced because the adsorbed polymer displaces the plane of shear. The parameters for describing adsorbed polymers are the fraction of the first layer covered by segments, 0, and the effective thickness, A, of the polymer layer, The insert gives the distribution of segments over trains and loops for polyvinyl alcohol adsorbed on silver iodide. Results obtained from double layer and electrophoresis measurements. 

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. 

The Stern surface is drawn through the ions that are assumed to be adsorbed on the charged wall. (This surface is also known as the inner Helmholtz plane [IHP], and the surface running parallel to the IHP, through the surface of shear (see Chapter 12) shown in Figure 11.9, is called the outer Helmholtz plane [OHP]. Notice that the diffuse part of the ionic cloud beyond the OHP is the diffuse double layer, which is also known as the Gouy-Chapman... 

As stated above, the results can be interpreted in terms of potential at the plane of shear, termed the zeta potential ( ). Since the exact location of the shear plane is generally not known, the zeta potential is usually taken to be approximately equal to the potential at the Stern plane. Two simple relations can be used to calculate zeta potentials in limiting cases, one for small particles with thick electric double layers, and one for large particles with thin electric double layers. 

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. 

In another part of this study we wished to see the effects of post-modification treatments on the properties of the modified LDPE surface. Polyethylene samples were photosulfonated for different periods of time. Afterwards they were subjected to an after-treatment by conditioning in an electrolyte solution (aqueous KC1, 10-3 M) for 48 hours and then characterized by zeta potential measurements. This conditioning process resulted in a shift of f to even less negative values (see Fig. 8). This finding may be explained by the swelling of the polymer samples (water adsorption) in water that causes a shift of the shear plane of the electrochemical double layer into the liquid phase. This effect demonstrates that storage conditions and pre-conditioning may exert a pronounced influence on the zeta potential recorded for surface-modified polymers. Phenomena of this kind have already been described in previous literature [26,27],... 

The size of the particles that is calculated from these experiments corresponds to particle dimensions plus the double layer thickness, in this case defined by the shear plane inside which the adsorbed species are rigidly held, and outside of which there is free movement. The shear plane can therefore be associated roughly with the outer Helmholtz plane, an approximation often made. The value of the electrostatic potential at the shear plane with respect to the value in bulk solution is called the electrokinetic or zeta potential, 33 (see Section 3.3). 

A double layer consists of a thin immobile Stern layer and a diffuse layer. The boundary between the two layers is known as the plane of shear. In this article, the particle surface is referred to as this plane. Thus, the surface charge density is contributed by the charge on the particle surface plus the charge in the Stern layer. In the diffuse layer, the flux of ionic species i is given by... 

Zeta potential — The electrical -> potential difference between the bulk solution and the shear plane or outer limit of the rigid part of the double layer (the limits of the diffuse - double layer) is the electrokinetic potential , often called the Zeta potential ((or more precisely the Zeta potential difference (). 

Figure 1 The electrical double layer and the potential distribution at the surface 1, fixed surface charges 2, Stern layer of "fixed" charges 3, Gouy or diffuse charge layer 4, counterions 5, Helmholtz plane 6, plane of shear 7, potential distribution in the electrical double 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. 

Recent surface force measurements revealed a similar trend (20). Comparing steam-treated to flame-treated silica sheets using site-dissociation/site-binding model, a decrease in silanol surface sites and apparent decrease in average pKa was observed upon heat treatment. Furthermore, a repulsive force other than double-layer and van der Waals forces was observed 15 A from the surface. This repulsion was attributed to hydration of the surface and was found to be independent of surface treatment and electrolyte concentration. In Bums treatment, an arbitrary plane of shear was introduced to provide a best model fit (l 3). A value of 9 A from the surface for the plane of shear was determined from electro-osmosis measurements. 

Shear Plane Any species undergoing electrophoretic motion moves with a certain immobile part of the electric double layer that is commonly assumed to be distinguished from the mobile part by a sharp plane, the shear plane. The zeta potential is the potential at the shear plane. 

Figure 9.19. The diffuse double layer, (a) Diffuseness results from thermal motion in solution, (b) Schematic representation of ion binding on an oxide surface on the basis of the surface complexation model, s is the specific surface area (m kg ). Braces refer to concentrations in mol kg . (c) The electric surface potential, falls off (simplified model) with distance from the surface. The decrease with distance is exponential when l/ < 25 mV. At a distance k the potential has dropped by a factor of 1/c. This distance can be used as a measure of the extension (thickness) of l e double layer (see equation 40c). At the plane of shear (moving particle) a zeta potential can be established with the help of electrophoretic mobility measurements, (d) Variation of charge distribution (concentration of positive and negative ions) with distance from the surface (Z is the charge of the ion), (e) The net excess charge.

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