An Ultra-Microporous Metal–Organic Framework with Exceptional Xe Capacity

Molecular confinement plays a significant effect on trapped gas and solvent molecules. A fundamental understanding of gas adsorption within the porous confinement provides information necessary to design a material with improved selectivity. In this regard, metal–organic framework (MOF) adsorbents are ideal candidate materials to study confinement effects for weakly interacting gas molecules, such as noble gases. Among the noble gases, xenon (Xe) has practical applications in the medical, automotive and aerospace industries. In this Communication, we report an ultra-microporous nickel-isonicotinate MOF with exceptional Xe uptake and selectivity compared to all benchmark MOF and porous organic cage materials. The selectivity arises because of the near perfect fit of the atomic Xe inside the porous confinement. Notably, at low partial pressure, the Ni–MOF interacts very strongly with Xe compared to the closely related Krypton gas (Kr) and more polarizable CO₂. Further ¹²⁹Xe NMR suggests a broad isotropic chemical shift due to the reduced motion as a result of confinement.     

On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001     

Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, homogeneous means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and longitudinal means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x – ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a longitudinal inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, inhomogeneous means that the wave is attenuated, in a direction distinct from the direction of propagation; and longitudinal means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = {S exp i(S · x – ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal longitudinal inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which longitudinal inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the ⁿ -ellipsoid, where is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.     

Inhomogeneous electromagnetic plane waves in crystals

Summary The propagation of inhomogeneous, time harmonic, elliptically polarised, electromagnetic plane waves in non-absorbing, magnetically isotropic, but electrically anisotropic, crystals is considered. The electric displacement and the magnetic induction are assumed to have the forms D exp l(S · x–t) and B exp l(S · x–t), respectively, at the place x and time t, where D, S, B are Gibbs bivectors (complex vectors) and is real. The implications of Maxwell's equations for the various field quantities are interpreted simply and directly through the use of bivectors and their associated ellipses.The propagation of circularly polarised waves is considered in detail. For such waves the electric displacement bivector is isotropic: D · D = 0. In order that such waves may propagate it is found that either (i) D is parallel to the slowness bivector S, so that both D and S are isotropic and coplanar, or (ii) D is parallel to the magnetic induction bivector B, so that both D and B are isotropic and coplanar. It is shown that for type (ii) the secular equation must have a double root for the slowness and conversely if the secular equation has a double root then there exists an isotropic electric displacement right eigenbivector of the optical tensor.Both types of waves are possible in a biaxial crystal. They complement each other in the following way. For type (i) all but two great circles on the unit sphere are possible circles of polarisation for circularly polarised waves with D and S parallel. Each of the exceptional great circles is such that an optic axis is normal to the plane of the circle. These two exceptional circles are the only possible circles of polarisation for circularly polarised waves of type (ii) when D and B are parallel.The situation for uniaxial crystals is similar—the only essential difference being that for uniaxial crystals there is only one exceptional circle since there is only one optic axis.For isotropic crystals the situation is quite different. Circularly polarised waves of type (i) are not possible. All great circles on the unit sphere are possible circles of polarisation for circularly polarised waves of type (ii) with D and B parallel.     

In General There Are No Unsheared Tetrads

Suppose the principal stretches are all different at a point P in a deformed body. In this case, it has been shown [1] that generally there is an infinity of non coplanar infinitesimal material line elements at P which remain unsheared following the deformation – that is, the angle between the arms of each pair of material line elements forming the triad remains unchanged. Here it is shown that in this case when all three principal stretches at P are different, there is no set of four infinitesimal material line elements, no three of which are coplanar, and such that the angle between each pair of the six pairs of material line elements is unchanged following the deformation. It is only when all three principal stretches at P are equal to each other, that there are unsheared tetrads at P, and in that case all tetrads are unsheared. This revised version was published online in June 2006 with corrections to the Cover Date.     

On energy flux and group velocity

Inhomogeneous plane wave solutions to the wave equations for a linear isotropic elastic solid and a linear isotropic dielectric are shown to possess energy flux velocity vectors which are non-coincident with corresponding group velocity vectors.In contrast to free surface waves, these examples imply a driving constraint and have an associated non-zero Lagrangian energy density.     

Connexions between the moduli for anistropic elastic materials

Summary For homogeneous isotropic elastic materials there are simple interrelations connecting Young's modulus, Poisson's ratio, the rigidity modulus and the modulus of compression. However for anisotropic materials the situation is quite different. Young's modulus is a function of direction and Poisson's ratio and the rigidity modulus are functions of pairs of orthogonal directions. Here some simple universal connexions between the moduli for various directions are simply derived for general anisotropic materials. No particular symmetry is assumed in the material.     

A study of two-dimensional dispersion in unconsolidated porous media

An experimental and numerical investigation into the magnitude of longitudinal and transverse dispersion in a two-dimensional flow field over a particle Peclet number range of 50–8500 is reported. Numerical modelling using a Galerkin finite element method is used to test various models, notably those of Fried and combarnous and Koch and Brady. Dispersion at low Peclet numbers (< 200) is found to be described adequately by either model, which at large Peclet, the degree of dispersion is significantly underestimated. An improved dispersion model for Peclet numbers greater than 200 is proposed. The transverse dispersion term and the choice of inlet boundary condition are found to have a negligible effect on the shape of the breakthrough curve.Nomenclature A (z) Polynomial in the z plane - B (z) Polynomial in the z plane - C Concentration - C _f Feed concentration - C _o Concentration at the entrance - D Dispersion tensor - D _f Molecular diffusion coefficient - D ₁ Longitudinal dispersion coefficient - D _p Particle diameter - D _t Transverse dispersion coefficient - k Permeability/viscosity - k Dimensionless permiability in the Koch and Brady model - P Pressure - Pe _k Modified Peclet number, Pe _p k - Pe _p Particle Peclet number vD _p /D _f - v Velocity - z Axial coordinate or complex variable Greek letters Solution domain - Boundary - Porosity     

Finite amplitude waves superimposed on pseudoplanar motions for Mooney-Rivlin viscoelastic solids

This paper deals exclusively with finite amplitude motions in viscoelastic materials for which the stress is the sum of a part corresponding to the classical Mooney-Rivlin incompressible isotropic elastic solid and of a dissipative part corresponding to the classical viscous incompressible fluid. Of particular interest is a finite pseudoplanar elliptical motion which is an exact solution of the equations of motion. Superposed on this motion is a finite shearing motion. An explicit exact solution is presented. It is seen that the basic pseudoplanar motion is stable with respect to the finite superposed shearing motion. Particular exact solutions are obtained for the classical neo-Hookean solid and also for the classical Navier-Stokes equations. Finally, it is noted that parallel results may be obtained for a basic pseudoplanar hyperbolic motion.