Plateau problem

W. Gozdz, R. Holyst. From the plateau problem to minimal surfaces in lipids, surfactants and diblock copolymer systems. Macromol Theory Simul 5 321-332, 1996. 

It can be shown that any solution to a Plateau problem must be a surface whose mean curvature H at each point is zero. Recall that H = + jcj],... 

T. Poston, The Plateau problem. An invitation to the whole of mathematics . Summer College on global analysis and its applications. 4 July-25 August 1972. International Centre for theoretical physics. Trieste, Italy 1972. 

More recently new mathematical techniques of differential geometry have been developed which required the use of currents, varifolds, and geometric measure theory. They are being applied with some success to the Plateau problem, the determination of the minimum area contained by a boundary. 

The mathematics of minimum surfaces is also being pursued with vigour in universities and research institutes. Mathematicians have been stimulated by the experiments of Joseph Plateau. In particular Jesse Douglas obtained the highest mathematical award, the Field s Medal, for his work on the Plateau Problem. In the future we can look forward to continued investigations into all aspects of surface properties. 

Polychloroprene consumption woddwide, except for eastern European countries and China, has plateaued at about 250,000 metric tons per year with some continued slow growth expected. Annual production averaged 307,000 metric tons during the 1980s with at least part of the difference being exported to formerly SociaUst countries. Production in Armenia has been limited to a fraction of its capacity of 60 metric tons by environmental problems and, in fact, is currendy shut down. The People s RepubHc of China has three plants with a combined capacity of 20 metric tons (2). 

The problem of concentration dependence of yield stress will be discussed in detail below. Here we only note that (as is shown in Figs 9 and 10) yield stress may change by a few decimal orders while elastic modulus changes only by several in the field of rubbery plateau and, moreover, mainly in the range of high concentrations of a filler. 

There are other sources of nonlinearity in the system, such as the intrinsic anharmonicity of the molecular interactions present also in the corresponding crystals. While these issues are of potential importance to other problems, such as the Griineisen parameter, expression (B.l) only considers the lowest order harmonic interactions and thus does not account for this nonlinear effect. We must note that if this nonlinearity is significant, it could contribute to the nonuniversality of the plateau, in addition to the variation in Tg/(do ratio. It would thus be helpful to conduct an experiment comparing the thermal expansion of different glasses and see whether there is any correlation with the plateau s location. 

An inadequate intake in the diet of those food chemicals that are essential nutrients results in health risks. Indeed these risks are by far the most important in terms of the world s population where malnutrition is a major public health problem. But, unlike the toxic chemicals, they would show a very different dose-response if they were subject to similar animal bioassays. At very low doses there would be a high risk of disease that would decrease as the dose was increased, the curve would then plateau until exposure was at such a level that toxicity could occur. Figure 11.2 shows this relationship which is U- or J-shaped rather than the essentially linear dose-response that is assumed for chemicals that are only toxic. The plateau region reflects what is commonly regarded as the homeostatic region where the cell is able to maintain its function and any excess nutrient is excreted, or mechanisms are induced that are completely reversible. 

Example.The kanamycin problem could be solved more aggressively as follows let Cmax = 35 pg/mL and Cmin = 10 pg/mL. From Eqs. (53) and (54), the value of R on the plateau may be calculated as follows ... 

Finally, a further unsolved problem should be mentioned. If we compare the plateau moduli of different polymer melts and relate them to the Kuhn length and to the density, this relation can also be adequately described with the scaling model, if an exponent a near 3 is chosen [73]. It is not known why this exponent is different if the contour length density is varied by dilution in concentrated solution or by selecting polymer chains of different volume. 

The problems associated with the application of this (or any other) model have been discussed. Because of the form of the typical isotherm, which exhibits a broad plateau region, fitting of experimental results to the model requires that data be obtained over a very broad range of concentrations. This is often very difficult to accomplish in practice, especially when difference methods are used to determine the amount of polymer adsorbed. Evaluation of adsorption in real systems is further complicated by a lack of knowledge of the available solid surface area. The latter may be affected by particle size, shape and surface topography and by polymer bridging between particles. 

At Coventry University [20] we have obtained similar results to Tu under conventional (silent) conditions. Our initial thoughts were that if the effect of the tail off was either a consequence of mass transfer to the electrode, or a consequence of some problem with the diffusion layer, then ultrasound might be expected to have an effect and thus improve the plating rate. Investigations in the presence of ultrasound and at various pH values did not significantly affect the plating characteristics i. e. the plateau effect still remained. However, the overall efficiency in the presence of ultrasound was affected (Fig. 6.10). 

Figure 5.24 shows that this approach fails not only quantitatively but also qualitatively. Neither is the strong increase of the collective times relative to the self-motion in the peak region of Spair(Q) explained (this is the quantitative failure) nor is the low Q plateau of tpair(Q) predicted (this is the quaUtative shortcoming). We note that for systems hke polymers an intrinsic problem arises when comparing the experimentally accessible timescales for self- and collective motions the pair correlation function involves correlations between all the nuclei in the deuterated sample and the self-correlation function relates only to the self-motion of the protons. As the self-motion of carbons is experimentally inaccessible (their incoherent cross section is 0), the self counterpart of the collective motion can never be measured. For PIB we observe that the self-correlation function from the protonated sample decays much faster than the pair... 

For simplicity, a linear relationship between concentration and effect is often assumed, reducing the problem of PK/PD to the pharmacokinetics. However, the concentration-effect relationship of any drug tends towards a plateau, and a sigmoidal model (sigmoid E ax model or Hill equation) is more appropriate [21-24] ... 

Neither a minimum nor a plateau is seen, but a random behaviour. This suggests problems with either the model or the data (too different samples, etcl [22]. 

That the Li abundance observed by Spite and Spite (1982) could not be the cosmological abundance but had been reduced by a factor of at least 4 by either diffusion or burning was first noted by Michaud, Fontaine and Beaudet (1984). These authors also emphasized that this plateau is constant over a surprisingly large Teff interval. This remains a problem requiring further study. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

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