Nodoids and toroids: comparison of two geometries for the meniscus profile of a wetting liquid between two touching isolated spheres and extensions to the model of a collection of packed spheres

In the mid 1960s the present authors published two papers dealing with penetration of nonwetting liquids such as mercury into the interstitial void spaces using the model of uniform packed spheres. A circular arc was used to approximate the liquid-vapor interface in both papers. However, our circular arc-toroid values for the pressure-volume relationship in the pendular ring which exists between two touching spheres was criticized. The authors concluded that our approximation led to unacceptably large differences compared to the values calculated from the exact nodoid shape. This incorrect conclusion was never rebutted and has, in fact, been misinterpreted by subsequent workers to include values calculated for the shape of the access opening and the associated pressure for penetration into the void space of a collection of spheres. This leaves a cloud of uncertainty, not only over our original work on nonwetting fluids, but on the application of our procedures to the field of wetting fluids. The contrast in the geometrical shapes of the toroid and nodoid is depicted and the pressure values are compared at equal volumes. In contrast to the claim of excessive error, we show the toroid geometry, in conjunction with a pressure-volume work derivation, to have a maximum error of 0.06% as compared to the nodoid at all liquid-solid contact angles. The toroid also has the advantage of using a readily derived work versus surface free energy balance rather than requiring the use of incomplete elliptic integrals to evaluate the nodoid. Attempts to use radii of curvature to evaluate the toroid shape are shown to give extremely poor approximations of the exact values for the pressure. Values reported for access to the interior void space of a collection of spheres still need adjustment for the effect of contact angles between 0 degrees and 180 degrees for characterizing assemblies of real solids by computing "equivalent spherical" particle size from porosity and mercury penetration data. However, there is no anticipation that use of the circular arc will introduce large errors in the results. This gives confidence to us and others working with wetting media to test the potential applicability of the packed sphere model to various diverse fields.     

Structure, electrochemistry, and magnetism of the iron(III)-substituted Keggin dimer, [Fe6(OH)3(A-alpha-GeW9O34(OH)3)2]11-

The iron(III)-substituted tungstogermanate [Fe6(OH)3(A-alpha-GeWO34(OH)3)2]11- (1) has been synthesized and characterized by IR, elemental analysis, SQUID magnetometry, electron paramagnetic resonance (EPR), and electrochemistry. Single-crystal X-ray analysis was carried out on Cs4Na7[Fe6(OH)3(A-alpha-GeW9O34(OH)3)2] x 30H2O, which crystallizes in the monoclinic system, space group C2/m, with a = 36.981(4) A, b = 16.5759(15) A, c = 16.0678(15) A, beta = 95.311(3) degrees, and Z = 4. Polyanion 1 consists of two (A-alpha-GeW9O34) Keggin moieties linked via six Fe3+ ions, leading to a double-sandwich structure. The equivalent iron centers represent a trigonal prismatic Fe6 fragment, resulting in virtual D3h symmetry for 1. Electrochemistry studies revealed that 1 is stable in solution from pH 3 to at least pH 7. In pH = 3 media the reduction of the six Fe3+ centers was featured by a single voltammetric wave for most supporting electrolytes used. In that case, whatever the scan rate from 1000 mV x s(-1) down to 2 mV x s(-1), no splitting of the single Fe-wave of 1 was observed. The acetate medium induced a partial splitting of the wave, and this separation is enhanced with increasing pH. Remarkable efficiency of 1 in the electrocatalytic reduction of nitrite, nitric oxide, and nitrate is demonstrated. Magnetic susceptibility (chi) measurements indicate a diamagnetic (S(T) = 0) ground state, with an average J = -12 cm(-1) and g = 2.00. EPR studies confirm that the ground state is indeed diamagnetic, since the EPR signal intensity steadily decreases without any line broadening as the temperature is lowered and becomes unobservable below about 50 K. The signal is a single broad peak at all frequencies (90-370 GHz), ascribed to the thermally accessible excited states. Its g(iso) is 1.992 51, as expected for a high-spin Fe3+-containing species, and supports the chi data analysis.     

Formation of transverse waves in oblique detonations

The structure of oblique detonation waves stabilized on a hypersonic wedge in mixtures characterized by a large activation energy is investigated via steady method of characteristics (MoC) calculations and unsteady computational flowfield simulations. The steady MoC solutions show that, after the transition from shock-induced combustion to an overdriven oblique detonation, the shock and reaction complex exhibit a spatial oscillation. The degree of overdrive required to suppress this oscillation was found to be nearly equal to the overdrive required to force a one-dimensional piston-driven detonation to be stable, demonstrating the equivalence of two-dimensional steady oblique detonations and one-dimensional unsteady detonations. Full unsteady computational simulations of the flowfield using an adaptive refinement scheme showed that these spatial oscillations are transient in nature, evolving in time into transverse waves on the leading shock front. The formation of left-running transverse waves (facing upstream) precedes the formation of right-running transverse waves (facing downstream). Both sets of waves are convected downstream away from the wedge in the supersonic flow behind the leading oblique front, such that the mechanism of instability must continuously generate new transverse waves from an initially uniform flow. Together, these waves define a cellular structure that is qualitatively similar to a normal propagating detonation.     

Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilizationof the wave equation

We deal here with a second order elliptic mixed problem which is posed in a regular open bounded domain of . We study the regularity of its solution. We apply our results to the boundary stabilization of the wave equation.     

Pressure-driven phase transitions in NaBH4: theory and experiments

We have investigated pressure-induced structural transitions in NaBH4 through density-functional theory calculations combined with X-ray and neutron diffraction experiments. Our calculations confirm that the cubic phase is stable up to 5.4 GPa and an orthorhombic phase occurs above 8.9 GPa, as observed in X-ray diffraction experiments. Both the calculations and X-ray diffraction measurements identify an intermediate tetragonal phase that appears between 6 and 8 GPa; that is, between the cubic and orthorhombic phases. This result is also confirmed by high-pressure neutron diffraction experiments performed on NaBD4. Our calculations and X-ray diffraction measurements show that the space group of the orthorhombic phase above 8.9 GPa is Pnma and the orthorhombic phase remains stable up to 30 GPa. The calculated equations of state are in excellent agreement with experiments.     

Photoionization of highly charged ions using an ECR ion source and undulator radiation

Photoionization of Xe4+ to Xe7+ ions was studied by combining an electron cyclotron resonance ion source with synchrotron radiation. Multiconfiguration Dirac-Fock calculations were performed to interpret the data. Many autoionization lines were measured and identified, resulting from excitation of a 4d electron into nf and np orbitals followed by Auger decay of the excited states. Continuum photoionization is negligible for the higher members of the isonuclear series.