Heterogeneous Sphere Research Articles

Cleavage of an aromatic carbon–heteroatom bond in a single step or successive steps?––A mechanistic distinction in the reduction of 5-bromo-1,3-dichloro-2-iodobenzene

The electrochemical reductive cleavage of the carbon–iodine bond in 5-bromo-1,3-dichloro-2-iodobenzene has been analyzed from the mechanistic point of view employing the Marcus theory of heterogeneous outer sphere electron transfer. The variation of the electrochemical transfer coefficient with potential has been estimated from the cyclic and convolution voltammetric studies. The comparison of the experimental reorganization energy with the theoretical predictions has revealed the existence of the radical anion of 5-bromo-1,3-dichloro-2-iodobenzene as a transient species.

Spatiotemporal response of a conducting sphere under simulated geomagnetic storm conditions

A theoretical study of satellite-based geomagnetic induction signals from uniform and heterogeneous spheres is undertaken using existing analytic and semi-analytic harmonic solutions. Transient responses to ring current excitations are computed using inverse Fourier and Laplace methods. A wide range of simulated storm excitations from impulse to step-on are considered. The early-time radial and late-time polar responses are diagnostic of the electrical conductivity of the sphere. The early-time diagnostic is likely more useful because of better signal to noise characteristics. Induction signals from deeply buried heterogeneities are likely to be contained in the late-time responses and their extraction will require new statistical approaches that average over many storms and incorporate multi-satellite and ground-based observations. The spatiotemporal induction response shows an interesting crossover during which time the rapidly-evolving polar component starts to decays while the slowly-evolving radial component continues to grow. At the crossover time, the intensity of the field is independent of colatitude, an effect which could be sought in satellite induction data.

A global and a local criterion in defining the tidal radius

AbstractThe basic theory of potential‐energy tensors related to heterogeneous spheres is reviewed, and the special case of truncated, singular, isothermal spheres is examined in detail. Special effort is devoted to a system made of two isothermal spheres, one completely lying within the other, the mass and the radius of the embedded sphere being negligible with respect to the mass and radius of the embedding sphere, and the radius of the embedded sphere being negligible with respect to the distance between the centres. The potential‐energy tensors related to the potential induced by the embedding sphere on the mass distribution of the embedded sphere, are expressed as the sum of two contributions: one, coming from the embedded sphere after collapse towards its centre, and one other, related to the actual mass distribution of the embedded sphere. Using the latter, both a global and a local criterion in defining the tidal radius of the embedded sphere, are formulated in connection with either the binding‐energy tensor or the virial‐energy tensor. In doing this, the tensor components along the axis joining the centre of the embedding and the embedded sphere, are considered. The global criterion is related to the whole, embedded sphere, while the local criterion is related to an infinitesimal mass element placed at the boundary of the embedded sphere, where the distance from the centre of the embedding sphere attains a maximum. The virial theorem in tensor form is splitted into two distinct expressions, related to orbital and intrinsic motions of the embedded sphere. Alternative criterions in defining the tidal radius of the embedded sphere, are formulated taking into consideration the centrifugal tensor potential and the tensor potential induced by orbital motions. With regard to a selected criterion, the tidal radius calculated with and without the inclusion of the centrifugal potential, exhibits a maximum variation by a factor of about two, related to circular orbits. An application is made, where the embedding and the embedded sphere are taken as representative of the Galaxy and a globular cluster, respectively. It is found that a stability region exists for both the global and the local criterion and, in addition, the instability first occurs at the perigalacticon, as expected in connection with instantaneous tidal radius. A powerlaw dependence of tidal radius from cluster mass and galactocentric distance, a*C ∝ M1/3CR2/30, is shown to be consistent with data from a sample of 16 objects investigated by Brosche et al. (1999). No significant correlation is found between the ratio of cluster radius to tidal radius and the orbital ratio of apogalacticon to perigalacticon, similar to averaged tidal radii defined by Brosche et al. (1999). An additional object, Pal 5, which is experiencing progressive disruption via tidal shocks during disk passages, is shown to be among the less bound (or more unbound) clusters, within the framework of the model. If the representation of globular clusters as isothermal spheres introduces only systematic errors in the ratio of cluster radius to tidal radius, aC/a*C = γ/η, then at least one other cluster, NGC5466 (which has the highest value between the sample objects), is inferred to undergo tidal disruption, in the model interpretation.

Relations between the zinc-blende and the NaCl structure type.

A new model for the transformation from the zinc-blende to the NaCl type is developed on the basis of the deformation of a 4-connected heterogeneous sphere packing in space group P3(2). The geometrical features of this transition are discussed by investigating the analogous deformations of a 4-connected homogeneous sphere packing in the supergroup P3(2)21 of P3(2). In this space group, the atomic arrangement corresponds to a diamond configuration. During the phase transition, it is deformed under preservation of all sphere contacts until a cubic primitive lattice is reached, the analogue of the NaCl arrangement formed by like atoms. The intermediate structure that occurs between the zinc-blende and the NaCl type is compared with the cinnabar type.

Solving elastodynamics in a fluid–solid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids

We present a spectral element approach for modeling elastic wave propagation in a solid–fluid sphere where the local effects of gravity are taken into account. The equations are discretized in terms of the displacement in the solid and the velocity potential in a neutrally stratified fluid. The spatial approximation is based upon a spherical mesh of hexahedra in which local refinement allows for adapting the discretization to the variation of elastic parameters in both the solid and the fluid regions. Continuity constraints across the non-conforming interfaces are introduced through Lagrange multipliers which are further discretized by the mortar element method. Due to the spherical nature of the non-conforming interfaces the mortar method turns out to be functionally conforming and allows for an equal-order interpolation of the primal variables and the Lagrange multipliers. The method is shown to provide an accurate solution when compared to analytical calculations obtained for radial models of elastic parameters. Its parallel implementation is based upon a simple domain decomposition strategy which makes it efficient to solve large problems as those imposed by planetary scales.

Nickel(II) hexamethyleneiminedithiocarbamate complexes with triphenylphosphine or tributylphosphine

A series of square-planar nickel(II) hexamethylenedithiocarbamate complexes with heterogeneous coordination spheres of composition [NiX(hmidtc)Y] · nCHCl3 [X = Cl, Br, I or NCS; hmi = C6H12, dtc = S2CN−; Y = PPh3 or PBu3, n = 0,1] have been synthesized and characterized by elemental analyses, i.r. and u.v.–v.i.s. spectroscopy, magnetochemical and conductivity measurements, and by thermal analysis. X-ray structures of [NiCl(hmidtc)(PPh3)] · CHCl3 and [NiBr(hmidtc)(PPh3)] · CHCl3 have been determined.

Effects of earth's spherical curvature and radial heterogeneity in dislocation studies—for a point dislocation

To the present, dislocation theories for a homogeneous half‐space are often used to calculate or interpret displacements and gravity changes caused by an earthquake or to inverse a seismic fault model. However, far‐field effects of spherical curvature and radial heterogeneity have to be considered. In this research, Okada [1985] and Sun et al. [1996] dislocation theories are used to calculate displacements caused by four independent dislocations in three earth models: a homogeneous half‐space, a homogeneous sphere, and a heterogeneous sphere. Effects of spherical curvature and radial heterogeneity are investigated through comparison of displacements. Results show that effects of both sphericity and stratification are very large. The stratified effect reaches a discrepancy of more than 25% everywhere on the surface of the earth, including the near field.

A proposal for a transition mechanism from the diamond to the lonsdaleite type.

A phase transition between the diamond (Fd3;m) and the lonsdaleite types (P6(3)/mmc) may be described as a deformation of a homogeneous sphere packing with three contacts per sphere (type 3/10/o1) in the common subgroup Pnna of Fd3;m and P6(3)/mmc. The frequently observed transition between the zinc-blende (F4;3m) and the wurtzite types (P6(3)mc) may be described in an analogous way as a deformation of a heterogeneous sphere packing in the subgroup Pna2(1). The proposed model guarantees the three-dimensional connection during the whole transformation process. By this property it is distinguished from other models.

On the transition from the wurtzite to the NaCl type.

A crystallographic interpretation of a possible transition mechanism from the wurtzite to the NaCl type is given. For this purpose, different atoms in the structures of both types are replaced by like ones, resulting in lonsdaleite-type configurations and in cubic primitive lattices, respectively. The atomic arrangements of both types correspond to homogeneous sphere packings. It is shown that a lonsdaleite configuration may be deformed into a cubic primitive lattice within the Wyckoff position Cmcm 8(f) m. 0,y,z. Such a transition is displacive since no bonds have to be broken. The corresponding phase transformation from the wurtzite to the NaCl type can be described as a deformation of a heterogeneous sphere packing. The intermediate structure would have the symmetry Cmc2(1).

Ionic Processes in Saturated Solutions of MgCl2·6H2O in Solutions of Other Divalent Metals

Solubility isotherms of MgCl2·6H2O at 25 °C in solutions of chlorides of bivalent beryllium, calcium, manganese, iron, cobalt, zinc and mercury were treated by the hydration analysis. In addition to the usual procedure, the recently developed modified transfer energy Ξ was used to characterize the situation in saturated solutions. As a result, the following ionic processes were established in the saturated solutions under consideration: competition for water in solutions of MgCl2·6H2O containing beryllium and calcium chloride, complex formation in solutions of MgCl2·6H2O containing manganese, cobalt, zinc and mercury ions and formation of ions with heterogeneous coordination sphere in solutions of MgCl2·6H2O containing iron or copper ions. In the last case, the ability of the formed ions to crystallize from solution was found to be reduced in comparison with the other cases. Activity coefficients of MgCl2 were calculated along the solubility curves under consideration.

A transition path from the zinc-blende to the NaCl type

In order to find a transition path from the zinc-blende to the NaCl type both structures are described with the aid of heterogeneous sphere packings. If all atoms in such crystal structures are replaced by like ones, atomic arrangements result that correspond to homogeneous sphere packings belonging to the diamond type or forming a cubic primitive lattice, respectively. It is shown, that a diamond configuration may be deformed into a cubic primitive lattice within the Wyckoff position Imma 4(e) mm2 0,¼,z. The corresponding phase transition in binary compounds from the zinc-blende to the NaCl type can be described as a deformation of a heterogeneous sphere packing in the subgroup Imm2 of Imma. Since no bonds have to be broken this type of transition is displacive. In addition, structural relations between high-pressure phases of semiconductors like silicon and germanium and related AB compounds are shown.

The NaCl- to CsCl-type phase transition discussed on the basis of the cP to cI deformation with the symmetry Cmcm 4(c) m2m

A structure forming a cubic primitive lattice cP may be deformed into a structure forming a cubic body-centred lattice cI in the space group Cmcm at position 4(c) m2m 0,y,1/4. If in related structures the sites are alternately occupied by unlike atoms, the NaCl and the CsCl types occur, respectively. The corresponding phase transition can be described as a deformation of a heterogeneous sphere packing in the subgroup Pmmn (a,-c,b) of Cmcm. All sphere configurations with symmetry Cmcm 4(c) m2m were derived. On the basis of this information, further possibilities for phase transitions that also correspond to sphere-packing deformations were found with this symmetry. Two of them possibly may take place in metals. The first one leads from a primitive hexagonal lattice to a hexagonal close packing, the other from a cubic body-centred lattice also to a hexagonal close packing.

Geoelectromagnetic induction in a heterogeneous sphere:a new three-dimensional forward solver using a conservative staggered-grid finite difference method

SUMMARY conservative staggered-grid finite diVerence method is presented for computing the electromagnetic induction response of an arbitrary heterogeneous conducting sphere by external current excitation. This method is appropriate as the forward solution for the problem of determining the electrical conductivity of the Earth’s deep interior. This solution in spherical geometry is derived from that originally presented by Mackie et al. (1994) for Cartesian geometry. The diVerence equations that we solve are second order in the magnetic field H, and are derived from the integral form of Maxwell’s equations on a staggered grid in spherical coordinates. The resulting matrix system of equations is sparse, symmetric, real everywhere except along the diagonal and illconditioned. The system is solved using the minimum residual conjugate gradient method with preconditioning by incomplete Cholesky decomposition of the diagonal sub-blocks of the coeYcient matrix. In order to ensure there is zero H divergence in the solution, corrections are made to the H field every few iterations. In order to validate the code, we compare our results against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model (Kuvshinov & Pankratov 1994), and against a quasi-analytic solution for an azimuthally asymmetric configuration of eccentrically nested spheres (Martinec 1998).

Exact theory of electromagnetic scattering by a heterogeneous multilayer sphere in the infinite-layer limit: effective-media approach

An exact theory is presented for calculating electromagnetic scattering by a model heterogeneous particle. The model is a multilayer sphere composed of alternating layers of different dielectric constants in the limit where the number of layers approaches infinity and the layer thickness approaches zero. The scattering analysis for this limiting case is based on a formulation of scattering theory known as the variable-phase method (VPM). A derivation of the VPM formulation is presented. Analytic formulas are derived for the scattering coefficients of the heterogeneous sphere that are generalizations of similar Mie theory formulas for scattering by a homogeneous sphere. The optical properties of the layered heterogeneous mixture are exactly described in this theory by the two effective-media parameters. The solution in the long-wavelength limit is used to derive a new formula for the effective dielectric constant for heterogeneous mixtures.

Normal mode oscillation of a sphere with solid–gas–solid structure

A free oscillation study was made for a heterogeneous sphere with three-layered spherical shell structure composed of an inner spherical sample, middle gas, and outer solid layers in order to apply resonant sphere technique (RST) to elasticity measurements at high pressures. The spheroidal modes of the sphere are classified into “sample,” “gas,” and “container” modes, whose strain energy distributions are localized in the sample, gas, and outer layers, respectively. The sample modes, whose frequencies provide information about the elastic constants of the spherical sample, are more strongly excited than the gas and container modes when putting a vibrational source on the surface of spherical sample in the three-layered sphere and measuring the resonance spectrum on the sample sphere. The localization of strain energy is due to a large difference in acoustic impedance between gas and solid layers. All the resonant frequencies increase with increasing pressure in the gas layer, but the pressure derivatives of resonant frequencies of gas modes are larger than those of sample and container modes. Thus the pressure curves of gas modes frequently intersect those of sample and container modes. At pressures where the sample or container modes intersect the gas modes, modal interchange takes place only between spheroidal modes of Sln and Sl′n′ satisfying the conditions of l=l′ and n−n′=±1. To avoid such an interchange, the gas layer should be kept thin.

Geomagnetic induction in a heterogeneous sphere: fully three-dimensional test computations and the response of a realistic distribution of oceans and continents

Long-period geomagnetic data can resolve large-scale 3-D mantle electrical conductivity heterogeneities which are indicators of physiochemical variations found in the Earth's dynamic mantle. A prerequisite for mapping such heterogeneity is the ability to model accurately electromagnetic induction in a heterogeneous sphere. A previously developed finite element method solution to the geomagnetic induction problem is validated against an analytic solution for a fully 3-D geometry: an off-axis spherical inclusion embedded in a uniform sphere. Geomagnetic induction is then modelled in a uniform spherical mantle overlain by a realistic distribution of oceanic and continental conductances. Our results indicate that the contrast in electrical conductivity between oceans and continents is not primarily responsible for the observed geographic variability of long-period geomagnetic data. In the absence of persistent high-wavenumber magnetospheric disturbances, this argues strongly for the existence of large-scale, high-contrast electrical conductivity heterogeneities in the mid-mantle. Lastly, for several periods the geomagnetic anomaly associated with a mid-mantle spherical inclusion is calculated. A high-contrast inclusion can be readily detected beneath the outer shell of oceans and continents. A comparison between observed and computed c responses suggests that the mid-mantle contains more than one order of magnitude of lateral variability in electrical conductivity, while the upper mantle contains at least two orders of magnitude of lateral variability in electrical conductivity.

N-Terminal extension of human immunodeficiency virus capsid protein converts the in vitro assembly phenotype from tubular to spherical particles.

Expression of retroviral Gag polyproteins is sufficient for morphogenesis of virus-like particles with a spherical immature protein shell. Proteolytic cleavage of Gag into the matrix (MA), capsid (CA), nucleocapsid (NC), and p6 domains (in the case of human immunodeficiency virus [HIV]) leads to condensation to the mature cone-shaped core. We have analyzed the formation of spherical or cylindrical particles on in vitro assembly of purified HIV proteins or inside Escherichia coli cells. CA protein alone yielded cylindrical particles, while all N-terminal extensions of CA abolished cylinder formation. Spherical particles with heterogeneous diameters or amorphous protein aggregates were observed instead. Extending CA by 5 amino acids was sufficient to convert the assembly phenotype to spherical particles. Sequences C-terminal of CA were not required for sphere formation. Proteolytic cleavage of N-terminally extended CA proteins prior to in vitro assembly led to the formation of cylindrical particles, while proteolysis of in vitro assembly products caused disruption of spheres but not formation of cylinders. In vitro assembly of CA and extended CA proteins in the presence of cyclophilin A (CypA) at a CA-to-CypA molar ratio of 10:1 yielded significantly longer cylinders and heterogeneous spheres, while higher concentrations of CypA completely disrupted particle formation. We conclude that the spherical shape of immature HIV particles is determined by the presence of an N-terminal extension on the CA domain and that core condensation during virion maturation requires the liberation of the N terminus of CA.

Pair Interactions between Heterogeneous Spheres

Surface heterogeneities on colloidal particles may cause nonuniform charge distributions, which result in electrostatic potential profiles differing from those seen when charge is distributed uniformly. This, in turn, can give rise to attractive interactions between pairs of such surfaces which may be of greater magnitude and range than van der Waals attractions. We extend previous work on interactions between pairs of heterogeneous plates to pairs of spheres, with nonuniform constant potential or constant charge boundary conditions. As in the planar case, large attractive free energies can be obtained which are not present in the equivalent uniform surfaces. Additionally, large restraining torques can be found, affecting the rotational motion. These torques are not present in uniform charge models and may go some way toward explaining restraining torques found in some experimental systems.

Pyrrolidinedithiocarbamate complexes of nickel with phosphorus donor ligands in the coordination sphere

The pyrrolidinedithiocarbamate nickel(II) complexes with heterogeneous coordination sphere of the composition [NiX(plddtc)(PPh 3)]·0.5CHC1 3 (X = Cl, Br; pld = C 4H 8, dtc = S 2CN −) and [Ni(plddtc) (dppe)]X · nH 2O (X = Cl, Br, I, NCS, NO 3, ClO 4, dppe = 1,2-bis(diphenylphosphino)ethane, n = 1, 2) have been synthesized. The compounds have been characterized by elemental analyses, IR and electron spectroscopies, thermal analysis, magnetochemical and conductivity measurements. A single crystal X-ray structure analysis has been used to study [NiBr(plddtc)(PPh 3)]·O.SCHC1 3. A deformed square-planar arrangement of donor atoms occurs in the NiS 2PBr chromophore.

Geomagnetic induction in eccentrically nested spheres

We present an analytic solution for a class of multidimensional forward problems of electromagnetic induction in a heterogeneous sphere. A variety of numerical methods for 3-D global electromagnetic modeling have been introduced recently, including this sheet, perturbation expansion, and finite element/finite difference schemes. The present analytic approach provides a model response against which the accuracy of more general numerical algorithms can be tested. We solve the electromagnetic induction problem in an azimuthally symmetric system of two eccentrically nested spheres and compute the surface magnetic vector potential A Φ and polar magnetic field B θ as functions of several of the model parameters.