Absorbed molecules in microporous hosts — Computational aspects

This review examines the computational tools that are available for modeling the behavior of absorbed molecules in microporous hosts such as the aluminosilicate zeolites. The role of quantum mechanical methods is briefly discussed, followed by a more detailed examination of approaches that utilize forcefields for both the zeolite host, itself, and the host-guest interactions. These approaches include molecular dynamics and Monte Carlo simulations. Several examples are presented, focusing on the behavior of benzene absorbed in zeolites belonging to the faujasite family. The development of an appropriate forcefield is described, and the utilization of this forcefield to study the structure, energetics, and dynamics of benzene in siliceous faujasite, Na-Y and sodium Na-X is explored. The results are compared with the available experimental data.     

Synthesis of d,l-β-carboxyaspartic acid from hydantoin-5-malonic acid diethyl ester

Hydantoin-5-malonic acid diethyl ester was synthesized by reduction of parabanic acid (oxalyl urea) to 5-hydroxy-hydantoin, conversion to 5-chlorohydantoin and condensation with malonic ester. Alkaline hydrolysis gave d,l-β-carboxyaspartic acid.     

Experimental isotherms of HCl on H2O ice under stratospheric conditions: connections between bulk and interfacial thermodynamics

The adsorption of HCl on the surface of H(2)O ice has been measured at temperatures and pressures relevant to the upper troposphere and lower stratosphere. The measured HCl surface coverage is found to be at least 100 times lower than currently assumed in models of chlorine catalyzed ozone destruction in cold regions of the upper atmosphere. Measurements were conducted in a closed system by simultaneous application of surface spectroscopy and gas phase mass spectrometry to fully characterize vapor/solid equilibrium. Surface adsorption is clearly distinguished from bulk liquid or solid phases. From 180 to 200 K, submonolayer adsorption of HCl is well described by a Bragg-Williams modified Langmuir model which includes the dissociation of HCl into H(+) and Cl(-) ions. Furthermore, adsorption is consistent with two distinct states on the ice substrate, one in which the ions only weakly adsorb on separate sites, and another where the ions adsorb as an H(+)-Cl(-) pair on a single site with adsorption energy comparable to the bulk trihydrate. The number of substrate H(2)O molecules per adsorption site is also consistent with the stoichiometry of bulk hydrates under these conditions. The ionic states exist in equilibrium, and the total adsorption energy is a function of the relative population of both states. These observations and model provide a quantitative connection between the thermodynamics of the bulk and interfacial phases of HCl/H(2)O, and represent a consistent physicochemical model of the equilibrium system.     

Hydrogenation of

The course of the hydrogenation of [5]- and [6]metacyclophane (1b and 1c) and their thermochemistry is described. Both compounds are hydrogenated rapidly (within 10 s) to furnish the bridgehead olefins 13b and 12c. The accompanying hydrogenation enthalpies are -220 and -141 kJmol(-1), respectively. Strain energies (SE) and olefinic strains (OS) of a number of bridgehead olefins have been evaluated by DFT calculations; it was concluded that 13b belongs to the class of hyperstable olefins which correlates nicely with its reluctance to undergo hydrogenation. By combining experimental hydrogenation enthalpies and DFT calculations, SE of 187 and 121 kJmol(-1) were derived for 1b and 1c.     

Multivariate data analysis for Raman imaging of a model pharmaceutical tablet

Spectroscopic imaging techniques provide spatial and spectral information about a sample simultaneously and are finding ever-increasing application in the pharmaceutical industry. Effective extraction of chemical information from imaging data sets is a crucial step during the application of imaging techniques. Multivariate imaging data analysis methods have been reported but few applications of these methods for pharmaceutical samples have been demonstrated. In this study, a bilayer model tablet consisting of avicel, lactose, sodium benzoate, magnesium stearate and red dye was prepared using custom press tooling, and Raman mapping data were collected from a 400 μm × 400 μm area of the tablet surface. Several representative multivariate methods were selected and used in the analysis of the data. Multivariate data analysis methods investigated include principal component analysis (PCA), cluster analysis, direct classical least squares (DCLS) and multivariate curve resolution (MCR). The relative merits and drawbacks of each technique for this application were evaluated. In addition, some practical issues associated with the use of these methods were addressed including data preprocessing, determination of the optimal number of clusters in cluster analysis and the optimization of window size in second derivative calculation.     

Engineering models for synthetic microvascular materials with interphase mass,momentum and energy transfer

New materials are being developed that consist of a solid matrix with pores or vessels through which a functional fluid phase may pass. The fluid can provide expanded functionality such as healing and remodeling, damage disclosure, enhanced heat transfer, and controlled deformation, stiffness and damping. This paper presents a class of engineering models for synthetic microvascular materials that have fluid passages much smaller than a characteristic structural length such as panel thickness. The materials are idealized as two-phase continua with a solid phase and a fluid phase occupying every volume. The model permits the solid and fluid phases to exchange mass, momentum and energy. Balance equations and the entropy inequality for general mixtures are taken from existing continuum mixture theory. These are augmented with certain definite types of solid–fluid interactions in order to enable adequately general, but workable, engineering analysis. The thermomechanical characteristics of this restricted class of materials are delineated. By demanding that the law of increase of entropy be satisfied for all processes, much is deduced about the acceptable forms of constitutive equations and internal state variable evolution equations. The paper concludes with a study of the uniaxial tension behavior of an idealized vascular material.     

Dependence of quasiliquid thickness on the liquid activity: a bulk thermodynamic theory of the interface

Studies of the phenomenon of quasiliquid formation span systems as diverse as noble gases, complex organic molecules, and metals, and span triple point temperatures from 25 to 933 K. We show that when viewed as a single phenomenon essentially all published measurements of the quasiliquid layer thickness on solids below the melting point can be plotted as a function of the thermodynamic activity. Two classes of behavior are then observed: one for molecular systems and one for atomic systems. We derive a dependence on activity through a grand canonical lattice gas calculation. This is the only such unifying theory of this phenomenon.     

Banach Space Model Theory II. Isomorphic Equivalence

Necessary and sufficient conditions are given for two Banach spaces to have linearly isomorphic ultrapowers. This is done firstly by carrying over the elementary equivalence of model theory to Banach spaces, and secondly by a game-theoretic approach basing on Ehren -feucht -Fraissé games from logic.     

Book Review

An Introduction to Difference Equations. Second Edition by Saber N. Elaydi, New York: Springer—Verlag, 1999. ISBN 0-387-98830-0. $54.95. Gone are the days when difference equations arose mainly in the context of sections of flows or as finite difference approximations to PDE's. Today difference equations have come into their own, both as objects of intrinsic mathematical interest and as dynamical models in their own right. Discrete models form an important part of dynamical systems theory independently from their continuous cousins. In Saber Elaydi's book dynamicists have the long awaited discrete counterpart to standard textbooks such as Hirsch and Smale (“Differential Equations, Dynamical Systems, and Linear Algebra”). The first edition of this book appeared in 1996. The second edition includes substantial new material including appendices on global stability and periodic solutions, a section on applications to mathematical biology, and a new chapter entitled “Applications to Continued Fractions and Orthogonal Polynomials”. Additional material on Birkhoff's theory now appears in the chapter on asymptotic behavior. The initial chapter covers first order equations, including equilibria, cobwebbing, stability, cycles, and the bifurcations of the discrete logistic equation. Chapter 2 moves on to higher order linear equations and briefly treats the difference calculus (for an in—depth treatment, see “Difference Equations: Theory and Applications. Second Edition” by Ronald E. Mickens, New York: Van Nostrand Reinhold, 1990). The subsequent chapters include systems of difference equations, stability theory, Z—transforms, control theory, oscillation theory, asymptotic behavior, and applications to continued fractions and orthogonal polynomials.

The chapters are composed of short sections, each of which ends with a nice selection of exercises. Answers to the odd—numbered problems appear in the back of the book. The core chapters include sections of applications to various fields such as population biology, economics, and physics. Several famous examples and topics are treated in the applications, including Gambler's Ruin, the Nicholson—Bailey host/parasitoid model, the heat equation, and Markov chains. Many discrete models are noninvertible, yet as many frustrated modelers know, most of the old standard treatments of linearization and the Stable Manifold Theorem., coming as they do from the context of sections of flows, require invertibility. Commendably, Elaydi avoids the needless assumption of invertibility in his stability theorems, and also in the Stable Manifold Theorem. However, invertibility is assumed in the Hartman—Grobman Theorem, where indeed it is necessary to establish conjugacy between the map and its linearization (see “An Introduction to Structured Population Dynamics”, CBMS—NSF Regional Conference Series in Applied Mathematics, Vol. 71, SIAM, Philadelphia, 1998 by J. M. Gushing, for an example of a noninvertible map for which the conjugacy fails. Readers may be interested to know that in this reference a weaker version of the Hartman—Grobman Theorem is proved that does not require invertibility but does establish the desired correspondence between types of hyperbolic equilibria in maps and their linearizations.)

This book is in Springer's Undergraduate Texts in Mathematics series and is indeed a very readable and appropriate text for advanced undergraduates or beginning graduate students. According to the author, the main prerequisites for such a course are calculus and linear algebra, with basic advanced calculus and complex analysis needed only for some topics in the later chapters. This is true; however in most situations the book would be best appreciated by students with a bit more mathematical maturity than is engendered by today's calculus and beginning linear algebra courses.Elaydi's book is a valuable reference for anyone who models discrete systems. It is so well written and well designed, and the content is so interesting to me, that I had a difficult time putting it down. I am pleased to own a copy for reference purposes, and am looking forward to using it to teach a senior topics course in difference equations.