THE ENERGY ASPECT OF THE RECIPROCAL INTERACTIONS OF PAIRS OF TWO DIFFERENT VIBRATION MODES OF A CLAMPED ANNULAR PLATE

The standardized mutual active and reactive sound power of a clamped plate, representing the energy aspect of the reciprocal interactions of two different in vacuo modes, has been computed. It was assumed that the vibrations are axisymmetric, elastic and time harmonic, the plate's transverse deflection is small as compared with the plate's size, and that the vibration velocity is small as compared with the acoustic wavenumber generated. The Kirchhoff-Love theory of a perfectly elastic plate was used. The integral formulae for the mutual sound power were transformed into their Hankel representations which made possible their subsequent computation. A closed path integral was used to express the integral in its Hankel representation to compute the mutual active sound power. The asymptotic stationary phase method was used to compute the two magnitudes, i.e., the mutual active and reactive sound power. The results obtained are the asymptotic formulae valid for the acoustically fast waves. The oscillating as well as the non-oscillating terms have been identified in the formulae to make possible their further separate analysis. The availability of the asymptotic formulae makes possible some fast numerical computations of the mutual sound power. Moreover, the formulae presented herein, together with those for the individual modes known from the literature, make a complete basis for further computations of the total sound power of the plate's damped and forced vibrations in fluid.     

Qy-excitation resonance Raman spectra of chlorophyll a and bacteriochlorophyll c/d aggregates. Effects of peripheral substituents on the low-frequency vibrational characteristics

Low-frequency (80-700 cm-1) Qy-excitation resonance Raman (RR) spectra are reported for thin-solid-film aggregates of several chlorophyll (Chl) a and bacteriochlorophyll (BChl) c/d pigments. The pigments include Chl a, pyrochlorophyll a (PChl a), methylpyrochloropyllide a (MPChl a), methylbacteriochloropyllide d (MBChl d), [E,M] BChl cS, [E,E] BChl cF, and [P,E] BChl cF. The BChl c/d's are the principal constituents of the chlorosomal light-harvesting apparatus of green photosynthetic bacteria. Together, the various Chl a's and BChl c/d's represent a series in which the peripheral substituent groups on the chlorin macrocycle are varied in systematic fashion. All of the Chl a and BChl c/d aggregates exhibit rich low-frequency vibrational patterns. In the case of the BChl c/d's, certain modes in the very low-frequency region (100-200 cm-1) experience exceptionally strong Raman intensity enhancements. The frequencies of these modes are qualitatively similar to those of oscillations observed in femtosecond optical experiments on chlorosomes. The RR data indicate that the low-frequency vibrations are best characterized as intramolecular out-of-plane deformations of the chlorin macrocycle rather than intermolecular modes. The coupling of the out-of-plane modes in turn implies that the Qy electronic transition(s) of the aggregate have out-of-plane character. The RR spectra of the BChl c/d's also reveal that the nature of the alkyl substituents at the 8 and 12 positions of the macrocycle plays an important role in determining the detailed features of the low-frequency vibrational patterns. The frequencies of the modes are particularly sensitive to larger substituent groups whose conformations may be more easily perturbed in the tightly packed aggregates. These factors also make aggregates of pigments containing larger substituents more susceptible to structural, electronic, and vibrational inhomgeneities. Collectively, the RR studies of the various pigments delineate the factors which influence the low-frequency vibrational characteristics of chlorosomal aggregates.     

Investigation of tightly coupled porphyrin arrays comprised of identical monomers for multibit information storage

Our prior designs for molecular-based information storage devices have employed multiple redox-active units organized in weakly coupled, covalently linked arrays. To explore a simpler design, we report here the synthesis of porphyrin arrays where porphyrins with identical oxidation potentials are directly linked to one another instead of joined via a molecular linker. Oxidative coupling with AgPF(6) of zinc(II)-5,15-bis(4-tert-butylphenyl)-10-phenylporphyrin, obtained by a rational synthesis, afforded the expected dimer joined by a meso-meso linkage and an unexpected trimer joined by meso-meso linkages. For attachment to an electroactive surface we synthesized a meso-linked porphyrin dimer with a thiol-linker in one of the meso positions. The S-acetyl protecting group was used to avoid handling free thiol groups. Coupling of zinc(II)-5,10,15-tris(3, 5-di-tert-butylphenyl)porphyrin ("upper half") and zinc(II)-5-[4-(S-acetylthio)phenyl]-10,20-bis(3, 5-di-tert-butylphenyl)porphyrin ("lower half") afforded three different meso-linked dimers with the desired dimer as the main product. Electrochemical examination of the meso-linked dimer in solution shows that the first two oxidation potentials of the array differ by approximately 0.15 V and straddle the value exhibited by the monomeric constituents. The third and fourth oxidation potentials of the array are also split although to a lesser extent ( approximately 0.08 V) than the first and second. For the meso-linked trimer, the first three oxidation waves are also split; however, these waves are severely overlapped. The electrochemical behavior of the dimers and trimer is indicative of strong electronic interactions among the porphyrins. The thiol-derivatized meso-linked dimers form self-assembled monolayers (SAMs) on gold via in situ cleavage of the S-acetylthio protecting group. The porphyrin SAM exhibits four well-resolved oxidation waves. Regardless, the meso-meso linkage is relatively unstable upon formation of the pi-cation radical(s). This characteristic indicates that the structural motif is of limited utility for molecular information storage elements.     

Swallowtail porphyrins: synthesis, characterization and incorporation into porphyrin dyads

The incorporation of symmetrically branched tridecyl ("swallowtail") substituents at the meso positions of porphyrins results in highly soluble building blocks. Synthetic routes have been investigated to obtain porphyrin building blocks bearing 1-4 swallowtail groups. Porphyrin dyads have been synthesized in which the zinc or free base (Fb) porphyrins are joined by a 4,4'-diphenylethyne linker and bear swallowtail (or n-pentyl) groups at the nonlinking meso positions. The swallowtail-substituted Zn(2)- and ZnFb-dyads are readily soluble in common organic solvents. Static absorption and fluorescence spectra and electrochemical data show that the presence of the swallowtail groups slightly raises the energy level of the filled a(2u)(pi) HOMO. EPR studies of the pi-cation radicals of the swallowtail porphyrins indicate that the torsional angle between the proton on the alkyl carbon and p-orbital on the meso carbon of the porphyrin is different from that of a porphyrin bearing linear pentyl groups. Regardless, the swallowtail substituents do not significantly affect the photophysical properties of the porphyrins or the electronic interactions between the porphyrins in the dyads. In particular, time-resolved spectroscopic studies indicate that facile excited-state energy transfer occurs in the ZnFb dyad, and EPR studies of the monocation radical of the Zn(2)-dyad show that interporphyrin ground-state hole transfer is rapid.     

Gårding's Theory of Hyperbolic Polynomials

This paper presents a simple, self‐contained account of Gårding's theory of hyperbolic polynomials, together with a recent convexity result of Bauschke‐Güler‐Lewis‐Sendov and an inequality of Gurvits. This account begins by establishing some new results. The first concerns the existence of a pointwise arrangement of the eigenvalues so that they become global real analytic functions. The second asserts that the associated “branches” are independent of the choice of hyperbolic direction. © 2013 Wiley Periodicals, Inc.