Planar Topological Defects in Unconventional Superconductors

In this work, we consider the properties of planar topological defects in unconventional superconductors. Specifically, we calculate microscopically the interaction energy of domain walls separating degenerate ground states in a chiral p-wave fermionic superfluid. The interaction is mediated by the quasiparticles experiencing Andreev scattering at the domain walls. As a by-product, we derive a useful general expression for the free energy of an arbitrary nonuniform texture of the order parameter in terms of the quasiparticle scattering matrix. The thesis is structured as follows. We begin with a historical review of the theories of superconductivity (Sec. 1.1), which led the way to the celebrated Bardeen-Cooper- Schrieffer (BCS) theory (Sec. 1.3). Then we proceed to the treatment of superconductors with so-called "unconventional pairing" in Sec. 1.4, and in Sec. 1.5 we introduce the specific case of chiral p-wave superconductivity. After introducing in Sec. 2 the domain wall (DW) model that will be considered throughout the work, we derive the Bogoliubov-de Gennes (BdG) equations in Sec. 3.1, which determine the quasiparticle excitation spectrum for a nonuniform superconductor. In this work, we use the semiclassical (Andreev) approximation, and solve the Andreev equations (which are a particular case of the BdG equations) in Sec. 4 to determine the quasiparticle spectrum for both the single- and two-DW textures. The Andreev equations are derived in Sec. 3.2, and the formal properties of the Andreev scattering coefficients are discussed in the following subsection. In Sec. 5, we use the transfer matrix method to relate the interaction energy of the DWs to the scattering matrix of the Bogoliubov quasiparticles. This facilitates the derivation of an analytical expression for the interaction energy between the two DWs in Sec. 5.3. Finally, to illustrate the general applicability our method, we apply it in Sec. 6 to the interaction between phase solitons in a two-band s-wave superconductor.

An investigation into the oxidation of organic sulphides, organic selenides and simple hydrocarbons by Mortierella isabellina NRRL 1757

The work presented in this thesis is divided into three separate sections 4!> Each' 'section is involved wi th a different problem, however all three are involved with a microbial oxidation of a substrate~ A series of 'aryl substituted phenyl a.nd be,nzyl methyl sulphides were oxidized to the corre~pondi~g sulphoxides by 'Mo:rtierellai's'a'b'e'llina NRR.L17'S7 @ For this enzymic Qxidation, based on 180 labeled experiments, the oxygen atom is derived fr'orn the atmosphere and not from water. By way of an u~.traviolet analysis, the rates of oxidation, in terms of sulphox~ de appearance, were obtained and correlated with the Hatnmett p s~grna constants for the phenyl methyl sulphide series. A value of -0.67 was obtained and, is interpreted in terms of a mechanism of oxidation that involves an electrophilic attack on the sulphide sulphur by an enzymic ironoxygen activated complex and the conversion of the resulti!lg sulphur cation to sulphoxide. A series of alkyl phenyl selen~des have been incubated with the fu~gi, Aspergillus niger ATCC9l42, Aspergillus fO'etidus NRRL 337, MIIJisabellina NF.RLl757 and'He'lminth'osparium sp'ecies NRRL 4671 @l These fu?gi have been reported to be capable of carrying out the efficient oxidation of sulphide to sulphoxide, but in no case was there any evidence to supp'ort the occurrence of a microbialox,idation. A more extensive inves·t~gation was carried out with'M,e 'i's'a'b'e'l'l'i'na, this fu~gus was capable of oxidizing the correspondi~g sulphides to sulphoxi.de·s·$ Usi:ng a 1abel.edsubstra.te, [Methyl-l4c]-methyl phenyl selenide, the fate of this compound was invest~gated followi!lg an i'ncubation wi th Me isabellina .. BeSUldes th. e l4C-ana1YS1Q S-,'. a quant"ltta"lve selen'lum ana1Y"S1S was carried out with phenyl methyl selenide. These techniques indicate that thesel'enium was capable of enteri!1g thefu!1gal cell ef'ficiently but that s'ome metabolic cleav~ge of the seleni'um-carbon bond' may take plac'e Ie The l3c NMR shifts were assigned to the synthesized alkyl phenyl sulphides and selenides@ The final section involved the incubation ofethylben~ zene and p-e:rtr.hyltoluene wi th'M ~ 'isab'e'llina NRRL 17574b Followi~ g this incubation an hydroxylated product was isolated from the medium. The lH NMR and mass spectral data identify the products as I-phenylethanol and p-methyl-l-phenylethanol. Employi!lg a ch'iral shift re~gent,tri~ (3-heptafluorobutyl-dcamphorato)'- europium III, the enantiomeric puri ty of these products was invest~gated. An optical rotation measurement of I-phenylethanol was in ~greement with the results obtained with the chiral shift re~gen,te 'M.isabe'l'lina is capable of carryi~g out an hydroxylation of ethylbenzene and p-ethyltoluene at the ~ position.

Biotransformation of polycylic aromatic compounds by fungi and an investigation into the oxidation of alkylbenzenes by Mortierella isabellina NRRL 1757

Incubations of several polycyclic heteroaromatic compounds and two polycyclic aromatic hydrocarbons with a series of common fungi have been performed. The fungi Cunninghamella elegans ATCC 26269, Rhizopus arrhizus ATCC 11145, and Mortierella isabellina NRRL 1757 were studied in this regard. Of the aza heteroaromatics, only dibenzopyrrole gave a ring hydroxylated product following the incubation with C. elegans. From the thio heteroaromatics studied, dibenzothiophene was metabolized by all the three fungi and thioxanthone by C. elegans and M. isabellina giving sulfones and sulphoxides. Thiochromanone was metabolized stereoselectively to the corresponding sulphoxide by C. elegans. Methyl substituted thioxanthones on incubation with C. elegans produced oxidative products, arising from S -oxidation and hydroxylation at the methyl group. Of the cyclic ketones studied, only fluorenone was reduced to hydroxyfluorene and this metabolism is compared with that reported with cytochrome P-450 monooxygenases of hepatic microsomes. A series of para-substituted ethylbenzenes has been transformed stereoselectively to the 1-phenylethanols by incubation with M. isabellina. Comparisons of the enantiomeric purities obtained from products with their respective para substituent of the same steric size but different electronic properties indicate that the stereoselectivity of hydroxylation at benzylic carbon may be susceptible to electron donating or withdrawing factors in some cases, but that observation is not va lid in all the comparisons. The stereochemistry of the reaction is discussed in terms of three possible steps, ethylbenzene ---) 1-phenylethanol ---) acetophenone ---) 1-phenylethanol. This metabolic pathway could account for the inconsistencies observed in the comparisons of optical purities and electronic character of para substituents. Furthermore, formation of 2-phenylethanol (in some cases), l-(p-acetylphenyl)ethanol from p-diethylbenzene, and N-acetylation of p-ethylaniline was observed. n-Propylbenzene was also converted to optically active 1-phenylpropanol. Acetophenone, p-ethylacetophenone, and o(,~,~-trifluoroacetophenone were transformed to 1-phenylethanol, l-(p-ethylphenyl)ethanol, and 1-phenyl-2,2,2-trifluoroethanol, respectively, with high chemical and excellent optical yields. The 13 C NMR spectra of several substrates and metabolic products have been reported and assigned for the first time.

The 21-dehydroxylation of corticosteroids: evidence for an enol intermediate and the hydroxylation of steroids by fungal cyto-chrome P450: evidence for a stepwise mechanism

Two enzyme mechanisms were examined: the 21-dehydroxylation of corticosteroids by the anaerobe Eubacterium l en tum, and the hydroxylation of steroids by fungal cytochrome P450. Deuterium labelling techniques were used to study the enzymic dehydroxylation. Corticosteroids doubly labelled (2H) at the C-21 position were incubated with a culture of Eubacterium lentum. It was found that t he enzymic dehydroxylation proceeded with the loss of one 2H f rom C-21 per molecule of substrate. The kinetic isotope ef fect f or the reaction was found to be k~kD = 2. 28. These results suggest that enzyme/substr ate binding in this case may proceed via t he enol form of the substrate. Also , it appears that this binding is, at least in part, the rate determining step of t he reaction. The hydroxylation of steroids by fungal cytochrome P450 was examined by means of a product study. Steroids with a double bond at the A8 (9), ~( lO ), or ~ (ll) position were synthesized. These steroids were then incubated with fungal strains known to use a cytochrome P450 monooxygenase to hydroxylate at positions allylic to these doubl e bonds. The products formed in these incubations indicated that the double bonds had migrated during allylic hydroxylat ion. This suggests that a carbon centred radical or ion may be an intermediate i n the cytochrome P450 cat alytic cycle.

The regulation of oat coleoptile phosphoenolpyruvate carboxylase and malic enzyme by Hâ ½ and metabolites : kinetic evidence for and against a cytosolic pH-stat

Phosphoenolpyruvate carboxylase (PEPC) and malic enzyme activities in soluble protein extracts of Avena coleoptiles were investigated to determine whether their kinetics were consistent with a role in cytosol pH regulation. Malic enzyme activity was specific for NADP+ and Mn2+. Maximal labelled product formation from [14C]-substrates required the presence of all coenzymes, cofactors and substrates. Plots of rate versus malate concentration, and linear transformations there- 2 of, indicated typical Michaelis-Menten kinetics at non-saturating malate levels and substrate inhibition at higher malate levels. pH increases between 6.5 and 7.25 increased near-optimal activity, decreased the degree of substrate inhibition and the Kmapp(Mn2+) but did not affect the Vmax or Kmapp(malate). Transformed data of PEPC activity demonstrated non-linear plots indicative of non-Michaelian kinetics. pH increases between 7.0 and 7.6 increased the Vmax and decreased the Km app (Mg2+) but did not affect the Kmapp(PEP). Various carboxylic acids and phosphorylated sugars inhibited PEPC and malic enzyme activities, and these effects decreased with pH increases. Metabolite inhibited malic enzyme activity was non-competitive and resulted mainly from Mn2+ chelation. In contrast, metabolite inhibited PEPC activity was unique for each compound tested, being variously dependent on the PEP concentration and the pH employed. These results indicate that fluctuations in pH and metabolite levels affect PEPC and malic enzyme activities similarly and that 3 the in vitro properties of PEPC are consistent with its proposed role in a pH-stat, whereas the in vitro properties of the malic enzyme cannot be interpreted in terms of a role in pH regulation.

Biotransformation of aromatic hydrocarbons and organic sulphides by fungi

Toluene is converted to benzyl alcohol by the fungi Mortierella isabellina and Helminthosporium species; in the latter case, the product is further metabolized. Toluene-a -d 1 , toluene-a,a-d2, and toluene-a,a,a-d 3 have been used with Mortierellaisabellina in a series of experiments to determine both primary and secondary deuterium kinetic isotope effects for the enzymic benzylic hydroxylation reaction. The values obtained, intermolecular primary kH/kD = intramolecular p rim a r y kH r kD = 1. 0 2 + O. 0 5, and sec 0 n dar y k H I kD = 1. 37 .:!. 0.05, suggest a mechanism for the reaction involving benzylic proton removal from a radical intermediate in a non-symmetrical transition state. 2H NMR (30.7 MHz) studies using ethylbenzene-l,1-d 2 , 3 -fluoroethylbenzene-l,1-d 2 , 4 -fluoroethylbenzene-l,1-d 2 , and toluene-dB as substrates with Mortierella isabellina suggest, based on the observable differences in rates of conversion between the substrates, that the hydroxylation of hydrocarbons at the benzylic position proceeds via a one electron abstraction from the aromatic ring, giving a radical cation. A series of 1,3-oxathiolanes (eight) were incubated with Mortierella isabellina , Helminthosporium , Rhizopus arrhizus , and Aspergillus niger . Sulphoxides were obtained from Mortierella isabellina and Rhizopus arrhizus using the substrates 2-phenyl-, 2-methyl-2-phenyl-, and 2-phenyl-2-tert. butyl-l,3-oxathiolane. The relative stereochemistry of 2-methyl-2-phenyl-l,3-oxathiolan-l-oxide was assigned based on lH decoupling, n.O.e, 1 and H NMR experiments. The lH NMR (200 MHz) of the methylene protons of 2-methyl-2-phenyl-l,3-oxathiolan-l-oxide was used as a diagnostic standard in assigning the relative stereochemistry of 2-phenyl-l,3-oxathiolan-l-oxide and 2-phenyl-2-tert. butyl-l,3-oxathiolan-l-oxide. The sulphoxides obtained were consistent with an oxidation occurring from the opposite side of the molecule to the phenyl substituent.

Kinetic and product studies for the oxidtion of organic sulfides and sulfoxides in the presence of transition metal complexes

Rates and products of the oxidation of diphenyl sulfide, phenyl methyl sulfide, p-chlorophenyl methyl sulfide and diphenyl sulfoxide have been determined. Oxidants included t-Bu02H alone, t-Bu02H plus molybdenum or vanadium catalysts and the molybdenum peroxo complex Mo0(02)2*HMPT. Reactions were chiefly carried out in ethanol at temperatures ranging from 20° to 65°C. Oxidation of diphenyl sulfide by t-Bu02H in absolute ethanol at 65°C followed second-order kinetics with k2 = 5.61 x 10 G M~1s"1, and yielded only diphenyl sulfoxide. The Mo(C0)g-catalyzed reaction gave both the sulfoxide and the sulfone with consecutive third-order kinetics. Rate = k3[Mo][t-Bu02H][Ph2S] + k^[Mo][t-Bu02H][Ph2S0], where log k3 = 12.62 - 18500/RT, and log k^ = 10.73 - 17400/RT. In the absence of diphenyl sulfide, diphenyl sulfoxide did not react with t-Bu02H plus molybdenum catalysts, but was oxidized by t-Bu02H-V0(acac)2. The uncatalyzed oxidation of phenyl methyl sulfide by t-Bu02H in absolute ethanol at 65°C gave a second-order rate constant, k = 3.48 x 10~"5 M^s""1. With added Mo(C0)g, the product was mainly phenyl methyl sulfoxide; Rate = k3[Mo][t-Bu02H][PhSCH3] where log k3 = 22.0 - 44500/RT. Both diphenyl sulfide and diphenyl sulfoxide react readily with the molybdenum peroxy complex, Mo0(02)2'HMPT in absolute ethanol at 35°C, yielding diphenyl sulfone. The observed features are mainly in agreement with the literature on metal ion-catalyzed oxidations of organic compounds by hydroperoxides. These indicate the formation of an active catalyst and the complexation of t-Bu02H with the catalyst. However, the relatively large difference between the activation energies for diphenyl sulfide and phenyl methyl sulfide, and the non-reactivity of diphenyl sulfoxide suggest the involvement of sulfide in the production of an active species.