Double-electric layer charge density

Eq. (3.92) implies a linear relation between cosh Y0 and cosh Ym. Therefore, values for go., and Fo, can be assessed by extrapolating this relation to cosh Ym = 1. This is demonstrated in Fig. 3.46 where a plot of cosh Yg versus cosh Ym is shown. Making use of least-squares analysis we obtain a slope of 1.01 0.01 and an intercept of 1.80 0.4. Furthermore, in conformity with Eq. (3.92) the intercept yields Fo, = 119 0.03 or, respectively, (po = 30.5 0.7 mV and <70i = 1.29 0.04 mC m 2 for the diffuse electric layer potential and diffuse double layer charge density at infinite separation. These values are in... 

Here, x is the distance to the charged surface and are, respectively, the number density and the potential energy (in kT units) of the ith ion in the double electric layer is the value of in the bulk solution the summation is carried out over all ionic species v is the average excluded volume per counterion and can be theoretically estimated as equal to eight times the volume of the hydrated counterion. 

The electrostatic stability of a colloidal system depends not only on the magnitude of the electrical surface charge density but also on the dielectric properties of the medium, on its ionic strength, on the valence of the ions in the double layer, on the size of the particles, and on the temperature of the system (only slightly). The total interaction potential between two spherical particles charged by a single type of ions at the surface can be determined using the DLVO equation ... 

In accord with Eqs. (IV.4) and (IV.5), the double-layer charge density and the electrical component of adhesion may be changed by changing the work function of one of the members of the contacting pair. It is usually easier to change the work function of the surface that is being exposed to dust. 

Let us examine the relationship between the electrical and molecular components of adhesive interaction. Such an analysis has been made starting from various premises. In [154], account was taken of the elastic properties of the contact surfaces in [151-153], the analysis was based on the relationship between the double-layer charge density in the contact zone and the adhesion... 

In detaching the particles, the force of adhesion is overcome. If the adhesion is determined by the electrical component, the adhesive force will be completely dependent on the double-layer charge density. Since each force of detachment corresponds to an adhesion number and a specific double-layer charge density, we can express the mathematical expectation for the electrical component of adhesion as follows ... 

The double-layer charge density og may assume its maximum value then the force of detachment will be equal to the electrical component of adhesive force, and the adhesion number will be a function of the maximum double-layer charge density, i.e., yp =f(of ). 

The arrangement of particles on a rough surface is random a particle may occupy any site relative to the peaks on the surface. Depending on the particle position, the double-layer charge density Og may vary, as well as the mathematical expectation of the electrical component of adhesive force Fg. 

But the situation will change if the air bubble surface will be filled with densely packed adsorption monolayer of ionized surfactants. In this case, the charge density of monolayer can reach 1 C/m. And the equal quantity of charges of opposite sign forms the diffuse part of the double electric layer. Due to this phenomenon, the flotation device allows to remove some ionic impurities from treated water, which contain sufficient amount of surfactants. And the exhaustion of surfactant at air bubble surface leads to the end of ion evacuation with bubble-film flow. The dissolved ions are removed due to the action of surfactants as cofactors of flotation. 

In the system of desalted water-air, when water doesn t contain any surfactants, the double electric layer at the air bubble surface is formed due to the structure and charge distribution in water molecules and due to the presence of charged hydroxyls and protons arisen as a result of dissociation of water molecules. In order to estimate the maximum charge density at the bubbles owing to water molecule dissociation, let s use the expression for the dissociation constant of water in its bulk and at the interface, respectively ... 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

Loeb, AL Overbeek, JTG Wiersema, PH, The Electrical Double Layer Around a Spherical Colloid Particle, Computation of the Potential, Charge Density, and Free Energy of the Electrical Double Layer Around a sperical Colloid Particle M.I.T. Press Cambridge, MA, 1961. Lorentz, HA, Wied, Ann. 11, 70, 1880. 

Studies of theadsorption of surface-active electrolytes at the oil,water interface provide a convenient method for testing electrical double-layer theory and for determining the state of water and ions in the neighborhood of an interface. The change in the surface amount of the large ions modifies the surface charge density. For instance, a surface ionic area of 100 per ion corresponds to 16 pC per square centimeter. " " ... 

According to the model proposed by Verwey and Niessen (1939), an electric double layer is formed at an ITIES, which consists of two ionic space charge regions. As a whole the electric double layer is electrically neutral, so for the excess charge density in the part of the double layer in the aqueous phase, and for the part in the organic phase,... 

This chapter is devoted to the behavior of double layers and inclusion-free membranes. Section II treats two simple models, the elastic dimer and the elastic capacitor. They help to demonstrate the origin of electroelastic instabilities. Section III considers electrochemical interfaces. We discuss theoretical predictions of negative capacitance and how they may be related to reality. For this purpose we introduce three sorts of electrical control and show that this anomaly is most likely to arise in models which assume that the charge density on the electrode is uniform and can be controlled. This real applications only the total charge or the applied voltage can be fixed. We then show that predictions of C < 0 under a-control may indicate that in reality the symmetry breaks. Such interfaces undergo a transition to a nonuniform state the initial uniformity assumption is erroneous. Most... 

Of the quantities connected with the electrical double layer, the interfacial tension y, the potential of the electrocapillary maximum Epzc, the differential capacity C of the double layer and the surface charge density q(m) can be measured directly. The latter quantity can be measured only in extremely pure solutions. The great majority of measurements has been carried out at mercury electrodes. 

Since the sorbing surface holds a charge, its electrical potential differs from that of the solution. The potential difference between surface and fluid is known as the surface potential T and can be expressed in volts. The product e 4 is the work required to bring an elementary charge e from the bulk solution to the sorbing surface. According to one of the main results of double layer theory, the surface potential is related to the surface charge density by,... 

The description of the sorption of charged molecules at a charged interface includes an electrostatic term, which is dependent upon the interfacial potential difference, Ai//(V). This term is in turn related to the surface charge density, electric double layer model. The surface charge density is calculated from the concentrations of charged molecules at the interface under the assumption that the membrane itself has a net zero charge, as is the case, for example, for membranes constructed from the zwitterionic lecithin. Moreover,... 

The Lippmann Equation. It can be shown thermodynamically that the slope of the electrocapillary curve is equal to the charge density, a, in the electric double layer (First Lippmann Equation). 

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