Thermal decomposition of a series of 1,2-diazetines

A homologous series of tricyclic diazetines (6a-c), differing by the number of methylene groups in the saturated bridges of the fused carbon bicycles, was synthesized. The DeltaH++ of decomposition for each of the diazetines to afford N2 and the corresponding alkene was determined experimentally: 6a, 31.7; 6b, 39.3; 6c, 38.8 kcal/mol. The ground-state strain energy of each diazetine was estimated utilizing computationally obtained DeltaHf's for each of the experimentally investigated diazetines as well as several other diazetines whose DeltaH++'s had been previously reported in the literature. The sum of the ground-state strain energies and DeltaH++'s of decomposition for all of the diazetines was nearly constant, with an average value of 59 kcal/mol, suggesting that all of the diazetines decompose via the same mechanism. Generally, the higher the ground-state strain energy of the diazetine, the less the DeltaH++ for decomposition. The decomposition transition states for 6a-c and 7 were modeled computationally at the RB3LYP/6-311+G(3df,2p)//UB3LYP/6-31+G(d,p) level. The agreement of the experimentally determined DeltaH++ values with transition-state energies obtained computationally supports the reaction mechanism originally proposed by Yamabe that the elimination process occurs by an unsymmetrical, yet concerted, transition state with strong biradical character.     

Expanding organoboron chemistry: epoxidation of potassium organotrifluoroborates

Most organoboron reagents are highly susceptible to oxidation by a variety of oxidants. In these processes, the valuable carbon-boron bond is cleaved. The organotrifluoroborates examined in this contribution appear resistant to many common oxidants. Consequently, thioether-substituted alkyltrifluoroborates can be oxidized to the corresponding sulfones, while retaining the trifluoroborate moiety. 1-Trifluoroboratoalkenes are oxidized by dioxirane, providing air-stable, crystalline oxiranyltrifluoroborates. These interesting materials are structurally analogous to oxiranyl anions, unstable intermediates that would otherwise be of tremendous synthetic utility. Finally, the first Suzuki-Miyaura coupling of an epoxytrifluoroborate has been accomplished. The chemistry described begins to expand organoboron chemistry in significant new directions.     

Application of isomorphous replacement in the structure determination of a cubic liquid crystal phase and location of counterions

A second generation monodendron with dodecyl end-groups based on the AB(3) monomer 3,4,5-trihydroxy benzoate has previously been shown to form a thermotropic cubic phase with Pmthremacr;n symmetry (Balagurusamy et al., J. Am. Chem. Soc. 1997, 119, 1539). A structure consisting of spherical "micelles" was proposed originally, but an alternative choice of structure factor phases, giving a structure of interlocked squashed columns, could not be ruled out by diffraction data on the original material alone. We have therefore synthesized two selectively fluorinated equivalent compounds, the carboxylic acid and its Rb salt, to be able to apply a variant of the isomorphous replacement crystallographic technique. On the basis of the electron density maps of the new labeled compounds, reconstructed using small-angle X-ray diffraction intensities, the interlocking columns model is unequivocally rejected and the spheres model is upheld. Furthermore, the location of the metal cation in the center of the "micelles" is directly confirmed. Micellar diameter was shown to decrease on fluorination of the dodecyl chain ends, and increase significantly on introduction of Rb. This is interpreted in terms of changes in the number of wedge-shaped dendrons fitting into a spherical micelle due to their changing taper angle. It was found that the Rb-rich regions at the centers of six out of eight "micelles" in the unit cell are elongated in the direction of their closest packing. This adds support to the suggestion of a partial "column-like" character of stacked rows of such micelles, consistent with the position of the Pmthremacr;n phase next to the columnar phase in the phase sequence of most taper-shape compounds. The results illustrate the potential of isomorphous replacement, used a great deal in protein crystallography, in structure investigation of liquid crystals and supramolecular soft matter.     

Evaluation of the two-dimensional reversed-phase-reversed-phase separations of low-molecular mass polystyrenes

The resolving power of four reversed-phase-reversed-phase two-dimensional (2D) chromatographic systems was evaluated using information theory (IT) and a geometric approach to factor analysis. The first separation dimension employed a C18 column, while the second separation dimension employed a carbon clad zirconia column. Mobile phases in each dimension were either methanol or acetonitrile. The sample matrix that was employed in this study contained a mixture of 58 components, comprised of stereoisomers and structural isomers of a systematic variation in molecular weight. Each of the components were oligostyrenes, with between two and five configurational repeating units having either n-butyl, sec-butyl or tert-butyl end groups. In the two-dimensional systems employed in this study, between 46 and 49 of the 58 components could be separated, depending on the mobile phase combinations--with apparent resolution that could not possibly be achieved in a single one-dimensional separation. The results from this study indicate that in order to fully evaluate the resolving power of a 2D system multiple methods of analysis that evaluate the separation potential are most appropriate. This becomes increasingly important when the sample contains components that are very closely related and the retention of solutes displays a high degree of solute crowding.     

The Forcing Convexity Number of a Graph

For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K ₂ for a nontrivial connected graph H is studied.     

Spin dependent fragmentation functions for heavy flavor baryons and single heavy hyperon polarization

Spin dependent fragmentation functions for heavy flavor quarks to fragment into heavy baryons are carculated in a quark-diquark model. The production of intermediate spin 1/2 and 3/2 excited states is explicity included. Λ _b , Λ_c and Ξ _c production rate and polarization at LEP energies are calculated and, where possible, compared with experiment. A different approach, also relying on a heavy quark-diquark model, is proposed for the small momentum transfer inclusive production of polarized heavy flavor hyperons. The predicted Λ_c polarization is roughly in agreement with experiment.     

Distance in stratified graphs

A graph G is stratified if its vertex set is partitioned into classes, called strata. If there are k strata, then G is k-stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color X in a stratified graph G, the X-eccentricity e _X(v) of a vertex v of G is the distance between v and an X-colored vertex furthest from v. The minimum X-eccentricity among the vertices of G is the X-radius rad_X G of G and the maximum X-eccentricity is the X-diameter diam_X G. It is shown that for every three positive integers a, b and k with ab, there exist a k-stratified graph G with rad_X G = a and diam_X G = b. The number s _X denotes the minimum X-eccetricity among the X-colored vertices of G. It is shown that for every integer t with rad_X G t diam_X G, there exist at least one vertex v with e _X(v) = t; while if rad_X G t s _X, then there are at least two such vertices. The X-center C _X(G) is the subgraph induced by those vertices v with e _X(v) = rad_X G and the X-periphery P _X (G) is the subgraph induced by those vertices v with e _X(G) = diam_X G. It is shown that for k-stratified graphs H ₁, H ₂,..., H _k with colors X ₁, X ₂,..., X _k and a positive integer n, there exists a k-stratified graph G such that C X _i(G) H _i (1 ; i ; k1) and for i j. Those k-stratified graphs that are peripheries of k-stratified graphs are characterized. Other distance-related topics in stratified graphs are also discussed.     

Diffusion of 111Cd probes in Ga7Pt3 studied via nuclear quadrupole relaxation

Diffusion of ¹¹¹In/Cd impurity atoms was studied in Ga₇Pt₃ at the atomic level using the method of perturbed angular correlation of gamma rays (PAC). As in previous measurements on Ga₇Pd₃, probes were observed to occupy two inequivalent Ga-sublattices. Quadrupole interaction perturbation functions for each site exhibited damping at elevated temperatures that is attributed to nuclear relaxation caused by stochastic jumps of the probe atoms accompanied by reorientation of axes of the electric field gradient tensor. Fitted relaxation frequencies, proportional to mean jump frequencies of the probe, were thermally activated. Arrhenius plots revealed jump-frequency activation enthalpies of 0.94 (8) and 0.67 (4) eV for the two sites and frequency prefactors close to 1 THz. Results were similar to those found previously for Ga₇Pd₃, although jump frequencies are about 100 times lower in Ga₇Pt₃.