Schwartzs Method

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

We start with the simplest problem of the plate-plate interaction. Consider two parallel plates 1 and 2 in an electrolyte solution, having constant surface potentials i/ oi and J/o2, separated at a distance H between their surfaces (Fig. 14.1). We take an x-axis perpendicular to the plates with its origin 0 at the surface of one plate so that the region 0 x h corresponds to the solution phase. We derive the potential distribution for the region between the plates (0 x h) on the basis of Schwartz s method [1]. The linearized Poisson-Boltzmann equation in the one-dimensional case is 

Biophysical Chemistry of Biointerfaces By HiroyuM Ohshima Copyright 2010 by John Wiley Sons, Inc. 

FIGURE 14.1 Interaction between two parallel dissimilar hard plates 1 and 2 at separation h. 

As the zeroth-order approximate solutions il/f x) and 1/ 2° (x), we choose the unperturbed potentials produced by plates 1 and 2 in the absence of interaction (i.e., when they are isolated), which are (see Chapter 1) 

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Wang Y A and Carter E A 2000 Orbital-free kinetic-energy density functional theory Theoretical Methods in Condensed Phase Chemistry (Progress in Theoretical Chemistry and Physics Series) ed S D Schwartz (Boston Kluwer) pp 117-84... 

One concerned with the measurement of gas-hquid interfacial tension should consult the useful reviews of methods prepared by Harkius [in Chap. 9 of Weissberger, Techniques of Organic Chemstry, 2d ed., vol. I, part 2, Interscience, New York, 1949), Schwartz and coauthors [Suiface Active Agents, vol. I, Interscience, New York, 1949, pp. 263-271 Suiface Active Agents and Detergents, vol. 2, Interscience, New York, 1958, pp. 389—391, 417—418], and by Adamson [Physical Chemistry of Suifaces, Interscience, New York, I960]. 

C. Schwartz, in Methods in Computational Physics, vol.2, ed. B. Alder, S. Fernback and M. Rotenberg (Academic Press, New York, 1963). 

L. Lederman (Batavia, Illinois), M. Schwartz (Mountain View, California) and J. Steinberger (Geneva) for the neutrino beam method and the demonstration of the doublet structure of the leptons through the discovery of the muon neutrino. 

Schwartz, C. [1963] Estimating Convergence Rates of Variational Calculations , Methods in Computational Physics, 2, p. 241. 

Wang, A., Carter, E.A., In Theoretical Methods in Condensed Phase Chemistry, Schwartz, S.D. (ed.), Dordrecht Kluwer, 2000, pp. 117-184. 

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

Schwartz LM, Osborne BA (eds) (1995) Methods in cell biology, vol 46. Academic Press, San Diego, CA... 

Epoxyfarnesol was first prepared by van Tamelen, Stomi, Hessler, and Schwartz 4 using essentially this procedure. It is based on the findings of van Tamelen and Curphey5 that N-bromosuccinimide in a polar solvent was a considerably more selective oxidant than others they tried. This method has been applied to produce terminally epoxidized mono-, sesqui-, di-, and triterpene systems for biosynthetic studies and bioorganic synthesis.6 It has also been applied successfully in a simple synthesis of tritium-labeled squalene [2,6,10,14,18,22-Tetracosahexaene, 2,6,10,15,19,23-hexamethyl-, (all-E)-] and squalene-2,3-oxide [Oxirane, 2,2-dimethyl-3-(3,7,12,16,20-pentamethyl-3,7,ll,-15,19-heneicosapentaenyl)-, (all-E)-],7 and in the synthesis of Cecropia juvenile hormone.8... 

Owing to the constraints, no direct solution exists and we must use iterative methods to obtain the solution. It is possible to use bound constrained version of optimization algorithms such as conjugate gradients or limited memory variable metric methods (Schwartz and Polak, 1997 Thiebaut, 2002) but multiplicative methods have also been derived to enforce non-negativity and deserve particular mention because they are widely used RLA (Richardson, 1972 Lucy, 1974) for Poissonian noise and ISRA (Daube-Witherspoon and Muehllehner, 1986) for Gaussian noise. 

Schwartz, A., Polak, E., 1997, Family of projected descent methods for optimization problems with simple bounds. Journal of Optimization Theory and Applications 92, 1... 

In this method, one alkyne is treated with Schwartz s reagent (see 15-17) to produce a vinylic zirconium intermediate. Addition of MeLi or MeMgBr, followed by the second alkyne, gives another intermediate, which, when treated with aqueous acid, gives the diene in moderate-to-good yields. The stereoisomer shown is the one formed in usually close to 100% purity. If the second intermediate is treated with I2 instead of aqueous acid, the 1,4-diiodo-1,3-diene is obtained instead, in comparable yield and isomeric purity. This reaction can also be done intramolecularly Diynes 56 can be cyclized to ( , E) exocychc dienes 57 by treatment with a zirconium complex. 

Many other authors, as reviewed extensively by Schwartz and Lorenzo, and by Eder improved the C18 RP-HPLC methods that have been largely applied using similar but not exactly identical systems to separate and to quantify complex mixtures of chlorophylls and carotenoids. 

Since the initial reports of Schwartz s reagent ([Cp2Zr(H)Cl]n, 1) over 30 years ago there has been explosive growth of zirconocene chemistry. Hydrozirconation is an efficient method of hydrometalation because of (i) the mild nature of the conditions involved, (ii) the excellent regio- and stereo-chemical control of hydrozirconation, (iii) the one-pot nature of the procedure, and (iv) the price of Zr, which is one of the... 

Although the Lagrangian method was able to handle several responses or dependent variables, it was generally limited to two independent variables. A search method of optimization was also applied to a pharmaceutical system and was reported by Schwartz et al. [17], It takes five independent variables into... 

Kachman, M. T. Wang, H. Schwartz, D. R. Cho, K. R. Lubman, D. M. A 2-D liquid separations/mass mapping method for interlysate comparison of ovarian cancers. Anal. Chem. 2002,74,1779-1791. 

Schwartz, J.L. 1993. In vitro biological methods for determination of carotenoid activity. Methods Enzymol 214 226-256. 

According to Spadaro and Rabl [41], damage costs of IQ decrement is likely the dominant part of the total damage costs of Pb. The dose-response function has been quite well characterized for Pb, for example by Schwartz [48] in a meta-analysis, who found that the IQ decrement is 0.026 IQ points for a 1 pg/L increase of Pb in blood. Spadaro and Rabl identified two possible ways of linking blood levels of lead to exposure. One of the methods connects incremental exposure of Pb in air to... 

Schwartz s Reagent3 is available commercially (from the Aldrich Chemical Company, Inc.) although it is quite expensive. Two literature preparations of this important reagent are available. The first utilizes LiAI(OtBu)3H to reduce zirconocene dichloride.4 The second method utilizes sodium bis(2-methoxyethoxy) aluminum hydride (RED-AL) as the reducing agent.2a The disadvantages of these procedures have been discussed.3... 

Schwartz, R. W. 1989. Chemical processing of PbTi03 by co-precipitation and sol-gel methods the role of powder and gel characteristics on crystallization behavior. PhD dissertation. University of Illinois at Urbana-Champaign. 

To understand these processes and correlate residue profiles with specific toxic responses required congener-specific methods of analysis and complex statistical techniques (principal component analysis). Using these techniques, it was established that eggs of Forster s terns of two colonies differed significantly in PCB composition (Schwartz and Stalling 1991). Similar techniques were used to identify various PCB-contaminated populations of harbor seals (Phoca vitulina) in Denmark (Storr-Hansen and Spliid 1993). 

Schwartz, J. et al., The development of methods for immunophenotypic and lymphocyte function analyses for assessment of Southern sea otter (Enhydra lutris nereis) health, Vet. Immunol. Immunopathol., 104, 1, 2005. 

Hydrozirconation of alkynes with the Schwartz reagent, Cp2Zr(H)Cl (6), and subsequent methylation is also a general method (Eq. 2.5) [16]. 

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