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Filtration Pretreatment
Published in Michael J. Matteson, Clyde Orr, Filtration, 2017
The overall effect of the attractive and repulsive forces is best expressed in the form of potential energy/distance, i.e., U = U(r), diagrams. If the potential energy increases as the distance of approach decreases, this corresponds to a repulsive or colloidally stable situation and vice versa. Figure 4a shows a colloidally stable situation in which curve I is the electrical double-layer repulsion term, curve II the van der Waals attraction term, and curve III the resultant obtained by adding curves I and II. Because of the strongly repulsive nature of curve I it dominates the form of the resultant interparticle repulsion curve. In Fig. 4b the repulsive term is much reduced and the resultant is therefore attractive. This corresponds to a colloidally unstable or coagulating situation.
Soil Release by Textile Surfaces
Published in M. J. Schick, Surface Characteristics of Fibers and Textiles, 2017
H. T. Patterson, T. H. Grindstaff
Electrical charge effects on soil removal affect particularly the separation of particulate materials from fiber surfaces. In his review of washing theory Kling [82] proposed that such particles were “snapped in” or held in place on fibers by a minimum in the potential energy-distance from the fiber curve produced by the combination of the van der Waals-London attraction forces and the electrical forces of repulsion. He cites references to support that both fibers and soil particles have negative charges in water and these charges are strengthened by hydroxyl ions, multivalent anions, and anionic surfactants in washing solutions. The result of this strengthening of electrical repulsion facilitates particulate soil removal. To explain the functioning of nonionic surfactants, Kling reports the results of ultracentrifuge studies which showed that adhesion of soil-model particles to cellulose in water was markedly reduced by nonionic surfactants, although not as much as by anionic materials. Schott [111] and Cutler et al [112] have also published extensive reviews in this area of study.
Density Regression with Conditional Support Points
Published in Technometrics, 2022
We first review the basics of support points (Mak and Joseph 2018) which is the building block of our method. Support points are used to compact a continuous probability distribution into a set of representative points. They are obtained by optimizing a statistical potential measure called the energy distance (Székely and Rizzo 2004, 2013). Let F and G be two cumulative distribution functions on a nonempty domain with finite means. The energy distance between F and G is defined aswhere and are independent and identically distributed copies from F and G, respectively. An important property of the energy distance is that , and if and only if F = G. Let G = Fn be the empirical distribution function for . Support points are then defined as