Space charge theory, assumptions

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

It should be mentioned that in the approach with nonzero electric divergence, the photon mass is also related to the space charges in vacuo. Now, in the approach with a / 0, we have j = ctE but jeff = 0. Let us now assume j = aE and j 7 0, which means fs 0. In such a case, jo is assumed to be associated with p, where p is the charge density in vacuo. So, in such an approach one can think of the existence of a kind of space charge in vacuo that is to be considered to be associated to nonzero electric field divergence. This will result in a displacement current in vacuum similar to that measured by Bartlett and Corle [43]. The assumption of the existence of space charge in vacuo makes our theory not only fully relativistic but also helps us to understand gauge condition. In the conventional framework of Maxwell s equations... 

Since space charge boundary layers make quantitative treatment quite difficult, Cabrera and Mott (37) have attempted to circumvent this difficulty in their well-known theory of metal oxidation by making the following two simplifications, whose applicability should first be discussed since they are not always justified (1) It is assumed that the space charge effects can be neglected in layers which are not thicker than several hundred A. (2) The assumption Is implied, although not explicitly stated, that the concentration of defects within the thin oxide layer is constant. Simplification (1) is justified to a first approximation in the formation of poorly conducting n-type oxide layers but not in other cases. 

It is important to stress that the density of states shown in Fig. 50 is not a fit to theory it is merely a diflFerent representation of the DLTS spectra shown in Fig. 46 that removes the nonlineaiities inherent in the response of the space-charge capacitance due to thermal emission from states at different energy depths. These nonlinearities are negligible for crystals with JVj, N-j but large for a-Si H because of the large number of deep states. The derived density of states follows directly from Poisson s equation based on our assumptions and auxiliary measurements. The energy scale is the superposition of two thermal emission energy scales one from the thermal emission of... 

The difficulty obviously originates from the assumption that the ions are point charges and can approach the surface charge without any limit. The theory will consequently become insufficient as soon as a considerable part of the space charge should be present, according to the theory, within a distance of, say, 5 x cm from the surface. 

The conclusion is that the GC theory is not satisfactory, mainly due to the assumption of the ions being point charges. It was Stem (1924) who clearly recognized this fact and proposed a modification to overcome that limitation. The modified model is depicted in Figure 3.11 an additional plane, the Helmholtz plane. Stem plane, or 2 plane is defined, as the distance of closest approach to the surface. Ions cannot get closer to the surface, and the space between the plane and the surface, termed the inner layer, is hlled by solvent, which behaves as a dielectric. This model is interpreted, from an electrical point of view, as two capacitors in series one is the inner layer or Stern capacitance, C , and the other is the diffuse layer capacitance, Q. The series connection means that the inverse of the capacitance is additive ... 

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