Electrical Debye length

An important characteristic of plasma is that the free charges move in response to an electric field or charge, so as to neutralize or decrease its effect. Reduced to its smaUest components, the plasma electrons shield positive ionic charges from the rest of the plasma. The Debye length, given by the foUowing ... 

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). 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

The basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration of charge carriers is very low in semiconductors (see Section 2.4.1). For this reason and also because the permittivity of a semiconductor is limited, the semiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths of the order of 10 4-10 6cm. This layer is termed the space charge region in solid-state physics. 

For a range of potential in which the interfacial charge is relatively small, the reciprocal of the interfacial electric capacity, C, of metal electrodes has conventionally been represented by a Laurent series with respect to the Debye length L-o of aqueous solution as shown in Eqn. 5-25 [Schmickler, 1993] ... 

In this expression, e is the dielectric permittivity of the suspending medium, is the electric surface potential, and k is the inverse Debye length [17], defined as ... 

It is instructive to compare this to the capacitance of a plate capacitor o A/d. Here, A is the cross-sectional area and d is the separation between the two plates. We see that the electric double layer behaves like a plate capacitor, in which the distance between the plates is given by the Debye length The capacity of a double layer — that is the ability to store charge — rises with increasing salt concentration because the Debye length decreases. 

What are the correct values of the potentials In the metal the potential is the same everywhere and therefore 99 has one clearly defined value. In the electrolyte, the potential close to the surface depends on the distance. Directly at the surface it is different from the potential one Debye length away from it. Only at a large distance away from the surface is the potential constant. In contrast to the electric potential, the electroc/zmz caZpotential is the same everywhere in the liquid phase assuming that the system is in equilibrium. For this reason we use the potential and chemical potential far away from the interface. 

We can observe electro-osmosis directly with an optical microscope using liquids, which contain small, yet visible, particles as markers. Most measurements are made in capillaries. An electric field is tangentially applied and the quantity of liquid transported per unit time is measured (Fig. 5.13). Capillaries have typical diameters from 10 fim up to 1 mm. The diameter is thus much larger than the Debye length. Then the flow rate will change only close to a solid-liquid interface. Some Debye lengths away from the boundary, the flow rate is constant. Neglecting the thickness of the electric double layer, the liquid volume V transported per time is... 

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... 

Electrostatic forces, acting when the electric double layers of two drops overlap, play an important role. As mentioned above, oil drops are often negatively charged because anions dissolve in oil somewhat better than cations. Thus, the addition of salt increases the negative charge of the oil drops (thus their electrostatic repulsion). At the same time it reduces the Debye length and weakens the electrostatic force. For this reason, emulsion stability can exhibit a maximum depending on the salt concentration. 

Equation 14.1 contains derivatives of both the concentration and electrical potential, and requires both quantities to be known as a function of position in the membrane so as to predict the flux. To simplify matters, Goldman proposed the approximation that the electric field across the membrane be considered constant (i.e., the electric field is not affected by the presence of ions in the membrane). Under these conditions, the electrical potential gradient reduces to E=—d

applied potential and h is the membrane thickness. This approximation is appropriate for membranes that are relatively thin compared to the Debye length. This is not the case for the skin, even when heat-separated ( 100 pm) or dermatomed ( 0.5 mm). This approximation is also reasonable when the total ion concentrations on both sides of the membrane are equal. However, this is rarely the case in iontophoresis for which the applied ionic concentration is typically much less than that subdermally. 

Rh is the hydrodynamic radius of the analyte, k is the inverse of the Debye length, r is the viscosity of the separation buffer, e is the fundamental unit of charge, and ft is a function that describes the effect of the molecule (or particle) on the electric field and is defined between two limits (i) the Htickel limit,/ = 1 when k,Rh < 1 (when the hydrodynamic radius is lower than the Debye length) and (ii) the Helmholtz-Smoluchovski limit, fi= /2 when k,Rh > 10 (when the hydrodynamic radius is higher than the Debye length). Between the limits / is calculated from the following equation ... 

For the near wall solution, from the slipping plane to 3 Debye lengths from the wall, the solution is restricted to cases where there are no inertial effects within this region. In other words the left hand side of (5) goes to zero and we are left with an equation of the same form as the DC electro-osmosis equation but now with an AC electric field. The solution to this equation is... 

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