Depletion layer effects

Syneresis. This chapter began with consideration of the depletion layer effect. This phenomenon can be seen in coatings that contain large latices (>300 nm) not highly stabilized by surface-attached (hydroxy-ethyl)cellulose fragments (16), and is in part the problem observed in the last sections of Chapter 27. The phenomenon is not necessarily restricted to HEC-thickened formulations and depletion flocculation. In our studies, syneresis is observed in thickened aqueous solutions and in dispersed systems containing the model trimer associative thickener (Scheme II) it can be overcome by addition of conventional surfactants. Syneresis in HMHEC-thickened solutions is discussed in Chapter 19 in the absence of a dispersed phase. Syneresis is discussed in the following chapter where additives that substantially enhance low shear viscosities are added to inhibit syneresis. 

Depletion layer effects occur in associative thickener formulations when the latex is larger in size ( 500 nm) and not highly stabilized with surface (hydroxyethyl)cellulose fragments. Syneresis is also observed in simple aqueous solutions and in latex dispersions when the hydrophobicity of the associative thickener is high. 

However, there is a condition which must be fulfilled for the above analysis to be valid. It is that the shear rate at a given radius in the tube is a unique function radius. This will normally be so if the tube radius is large compared with the molecular dimension of the polymer. However, for very narrow capillaries, this may not be the case and the solution may become depleted in polymer molecules close to the capillary wall through the depleted layer effect (see Chapters 6 and 7). Thus, the concentration may vary across the capillary, and hence the constitutive model relating rj and y must also depend on local concentration and there is not a unique inversion of the rj/y relationship. This will be discussed in detail in Chapter 6, which will refer back to the development of the Mooney-Weissenberg-Rabinowitsch equations in this context (Sorbie, 1989, 1990). 

Figure 6.6. Schematic diagram of (a) polymer concentration profile, C(r), as a result of the depleted layer effect and (b) molecular origin of the effect as a result of entropic exclusion of molecules from near the wall region.

In Section 6.3.3 some of the experimental evidence for and theoretical interpretation of a depleted layer effect close to the pore wall was reviewed. This is discussed here in the context of a narrow cylindrical capillary in which the hydrodynamic problem can be formulated. Figure 6.7 shows that there is a polymer concentration profile, C(r), across the capillary and that the depleted layer extends over a distance S from the wall. Some notation is introduced in Figure 6.6 r is the radial co-ordinate, i 2 is the tube radius, is the radius of the inner bulk region, and the depleted layer thickness, d, is given by d = R2 — Ri). In order to perform calculations on the hydrodynamic effects which are caused by this profile, it is necessary to have a single analytic form for either the concentration profile, C(r), or the resulting viscosity profile, rj r), directly. The latter quantity must be derived in the calculations if it is not directly available. In the absence of a depleted layer. 

Sorbie, K. S. (1989) Network modelling of xanthan rheology in porous media in the presence of depleted layer effects. SPE 19651, Proceedings of the SPE 64th Annual Fall Conference, San Antonio, TX, 8-11 October 1989. 

Although excluded-volume effects are well-accepted in polymer chromatography, the preceding arguments are not without controversy in the petroleum literature because of the way that the apparent polymer viscosity in porous media was determined. In our work, we simply report the resistance factor (i.e., the brine mobility before polymer injection divided by the polymer-solution mobility). This is a well-defined parameter that derives directly from the Darcy equation and measurements of pressure drops and flow rates. Advocates of the depletion-layer effects use a different method to determine apparent polymer viscosity in porous media. Specifically, they flush water through the core after polymer injection to determine the permeability reduction or residual resistance factor. The resistance factor during polymer injection is then divided by the residual resistance factor to determine the apparent polymer viscosity in porous media. Unfortunately, several experimental factors can lead to incorrect measurement of high residual resistance factors, which, in turn, lead to calculation of unexpectedly low apparent polymer viscosities in porous media. 

Sorbie, K.S. 1989. Network Modeling ofXanthan Rheology in Porous Media in the Presence of Depleted Layer Effects. Paper SPE 19651 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 8-11 Pctober. DPI 10.2118/19651-MS. 

Otherwise, the effect of electrode potential and kinetic parameters as contained in the relevant expression for the PMC signal (21), which controls the lifetime of PMC transients (40), may lead to an erroneous interpretation of kinetic mechanisms. The fact that lifetime measurements of PMC transients largely match the pattern of PMC-potential curves, showing peaks in accumulation and depletion of the semiconductor electrode and a minimum at the flatband potential [Figs. 13, 16-18, 34, and 36(b)], demonstrates that kinetic constants are accessible via PMC transient measurements, as indicated by the simplified relation (40) derived for the depletion layer of an n-type electrode. 

More subtle effects of the dielectric constant and the applied bias can be found in the case of semiconductors and low-dimensionality systems, such as quantum wires and dots. For example, band bending due to the applied electric field can give rise to accumulation and depletion layers that change locally the electrostatic force. This force spectroscopy character has been shown by Gekhtman et al. in the case of Bi wires [38]. 

When Es > FB, region (b), a depletion layer forms in the semiconductor due to the bending of the bands under the influence of the electric field. Increasing the potential increases this band bending and so increases the effective barrier to tunnelling it represents. However, the high doping level... 

Eventually, in region (d), tunnelling occurs from the tip to the sample. Although the depletion layer is still thick, the effective thickness of the barrier in this region is actually reduced and the presence of the surface states plays a dominant role in maintaining the tunnelling current in this region. 

The working principle of LAPS resembles that of an ion-selective field effect transistor (ISFET). In both cases the ion concentration affects the surface potential and therefore the properties of the depletion layer. Many of the technologies developed for ISFETs, such as forming of ion-selective layers on the insulator surface, have been applied to LAPS without significant modification. 

The principle of depletion is illustrated in Figure 1. If a surface is in contact with a polymer solution of volume fraction , there is a depletion zone near the surface where the segment concentration is lower than in the bulk of the solution due to conformational entropy restrictions that are, for nonadsorbing polymers, not compensated by an adsorption energy. The effective thickness of the depletion layer is A. Below we will give a more precise definition for A. 

According to the macropore formation mechanisms, as discussed in Section 9.1, the pore wall thickness of PS films formed on p-type substrates is always less than twice the SCR width. The conductivity of such a macroporous silicon film is therefore sensitive to the width of the surface depletion layer, which itself depends on the type and density of the surface charges present. For n-type substrates the pore spacing may become much more than twice the SCR width. In the latter case and for macro PS films that have been heavily doped after electrochemical formation, the effect of the surface depletion layer becomes negligible and the conductivity is determined by the geometry of the sample only. The conductivity parallel to the pores is then the bulk conductivity of the substrate times 1 -p, where p is the porosity. 

The thickness of depletion and deep depletion layers may be approximated by the effective Debye length, Lo, ff, given in Eqn. 5-70 Ld, is inversely proportional to the square root of the impiuity concentration, In ordinary semiconductors... 

The difference between a photoconductive detector and a photodiode detector lies in the presence of a thin p-doped layer at the surface of the detector element, above the bulk n-type semiconductor. Holes accumulate in the p-layer, and electrons in the n-type bulk, so between the two there is a region with a reduced number density of carriers, known as the depletion layer. The important effect of this is that electron-hole pairs, generated by photon absorption within this depletion layer, are subjected to an internal electric field (without the application of an external bias voltage) and are automatically swept to the p and n regions, and... 

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