Michael Jacob

Hydrogen in carbon nanostructures

The non-dissociative hydrogen uptake for two types of carbon-based materials was explored. The maximum measured differential hydrogen uptake capacities at hydrogen pressures of 10 MPa were experimentally determined with the volumetric Sieverts method to 2.9 (300 K), and 4.1 wt% (77 K) for MWNB (multi-walled nanobarrels) and 2.6 (300 K), and 3.3 (77 K) wt% for ANPC (amorphous nanoporous carbon). The total hydrogen uptake was calculated to be 3.2 (300 K) and 6.2 (77 K) wt% for MWNB at a hydrogen pressure of 10 MPa, and 2.8 (300 K) and 4.2 (77 K), respectively, for ANPC. The adsorption energies were determined to be 7.2 and 4.2 (KJ/mol) for ANPC and MWNB, respectively, at the lowest coverage. At higher coverage (concentration), a multi-site model is required to describe the coverage dependence, consistent with large heterogeneity of the adsorption sites.

Bulk Synthesis of Nanotube-Like Carbon Material

Nanoporous carbons have been synthesised through a selective etching reaction, performed by halogenisation of metal carbides. The structures obtained can be controlled by choice of starting materials and reaction parameters, and here one example, chlorination of aluminium carbide at 700 °C, is given. The produced material is nanotube-like with agglomerates of short and wide interconnected tubular structures, which are here described as nanobarrels. The synthesis process gives a pure product with high-yield, and may be scaled up to produce bulk amounts.

The Natural Function and the Exponential Scale

This chapter examines the various aspects of the natural function and the exponential scale. The equations of symmetry in space and the fundamental polyhedral which includes the cube, the tetrahedron, the octahedron, and the rhombic dodecahedron are derived. The equations for the icosahedron and the pentagonal dodecahedron from the equations of fundamental polyhedral are presented. The expressions of the curvatures by looking at three special cases, a vertex, an edge, and the middle of a face are elaborated. At a vertex the coordinates are one of the eight permutations and the Gaussian curvature in such a point is analyzed. It is found that the Gaussian curvature converges to zero, since the edge is similar to a cylinder and has parabolic geometry. It is observed that the smaller the polyhedra, the lower the constant and the more each vertex gets affected by the others that results in the polyhedra turning spherical and the curvatures increase.

Handmade Structures and Periodicity

This chapter examines the various aspects of handmade structures and periodicity. It is necessary to go higher up in the exponential scale, in order to keep the original characters of the units put together into a continuous function. It is found that going to the exponential scale and adding the same two spheres with different centers with equation gives again complete fusion. Two different functions can be added on the exponential scale so that the sum function is continuous and the properties of the original functions are kept. It is observed that increasing the constant toward unity makes the planes come together and the geometry is approaching the topology for the pseudosphere, famous for having constant negative Gaussian curvature. The topology shows that many structures in nature might well be built with constant negative curvature. The mathematics of the hyperbolic plane is difficult and not possible to use directly. It is found that by starting to vary the constant, the gyroid surface is obtained with the typical monkey saddle.

The Screw and the Finite Periodicity with the Circular Functions

This chapter examines the space curves and the time parameterization. Finite periodicity from circular functions is introduced. A molecule can be left or right handed. Their physical properties are identical but a molecule used as a drug must have correct chirality to be active. With wrong chirality it might have no effect at all or it might be deadly poison. Translation and rotation are special cases of motions in space and the combination of the two give the screw, which is chiral. It is found that the structure of the helicoid is composed of the two simpler surfaces glued together. Squaring the equations for the building planes of the tower surfaces makes the saddles close up and give the so-called disc surfaces. It is observed that by adding a cylinder exponentially, one side of the surface will close up and result in the structure. This represents a DNA double-helix with 0 base pairs per pitch, displaying also the bridging hydrogen bonds created by the saddles. The bending of a helix is also elaborated.

The Rings, Addition and Subtraction

This chapter discusses the various aspects of the rings, addition, and subtraction. Some simple examples of subtraction and addition in 3D are presented. The whole cube and the sphere are subtracted and the sphere is distorted. The liberated cube is shown and calculated with the same function but with different boundaries. At higher constants there is catenoidic contact between the sphere and the cube. A slightly modified formula for the classic catenoid minimal surface is presented. It is found that the structure of the function is two parallel planes perpendicularly meeting a cylinder without self-intersections. Adding a sphere to the catenoid means closing it. Subtracting a sphere means bending the planes so they meet and become a torus. The pentagonal dodecahedron is subtracted from a sphere, and the resulting dual, the hyperbolic icosahedron is shown. A cube may be constructed from two dual tetrahedral. The result of the subtraction of a tetrahedron from a cube is also presented.

The Gauss Distribution Function

This chapter discusses the remarkable properties of the Gauss distribution (GD) function, and uses it to describe finite periodicity and the geometry of molecules, small ones or large ones, as cubosomes. Models for defects in crystals are also given. The broken symmetry in DNA and the possibility of a mathematical code matching are elaborated. It is found that more symmetry groups in space are obtained through the permutation variables, using the GD function. The boundary properties of the finite periodical function close the surface and form the particle. This is a possible Larsson cubosome of the G type or also a crystal with the structure of garnet. A model for different grooves in DNA is also given. By mixing phases the outer shape of a crystal can be varied. Nonconvex polyhedra are also shown. A structure of dilatation symmetry is given. The link to cosine is analyzed. The shape of several radiolarian creatures is also derived.

Multiplication, Nets and Planar Groups

This chapter describes various aspects of multiplication, nets, and planar groups. The mathematical equations for nets and the planar square groups and the quasiperiodic symmetry are derived. A nonzero constant gives a primitive structure of bodies. The cosine addition in sin x .sin y .sin z +cos x .cos y .cos z gives the diamond net and so on. It is found that with four planes or more, the use of irrational numbers from the general saddle equation brings out quasiperiodic symmetry. Five planes in the saddle way and with sine and a constant of 0.31, show a beautiful fivefold quasiperiodic symmetry, outside the origin. In analogy with observations, this should be a structure, and it remarkably agrees with the quasistructure model. It is observed that the four planes build two identical square nets that interpenetrate to an incommensurate structure. The fourfold symmetry shows up beautifully in a zeolite like structure. The symmetries are derived from the rotations of simple nets of translational periodicity.

The Rods in Space

This chapter describes the mathematics for cylinder packing in space. The geometry of condensing of packing into surfaces is presented. The Gauss distribution mathematics offers a way of deriving the rod packings. It is found that if one put a central goke in the primitive packing and made all the rods have the same diameter and touch each other, they all are space diagonals of the cube. The packing is bcc of rods. It is shown that in the tetragonal packing, the rods are parallel to face diagonals. The packings are demonstrated directly by using the cube. It is observed that blowing a thin beam of air onto the body along the threefold axis means spinning as the other rod system of opposite chirality is hidden. Organizing another beam of exactly opposite direction, blowing simultaneously, would make the body spin even better. It is found that the change of constant for the primitive and the garnet, or bcc, packing of rods, make the rods condense into surfaces of exactly the same type as the case for the circular functions.

The Roots of Mathematics—the Roots of Structure

This chapter describes the fundamental theorem of algebra in two and three dimensions. Polynomial products with suitable roots contain the commencement of periodicity. Permutations of variables and polynomial additions in three dimensions give the fundamental polyhedra, structure of simple molecules in natural science, and the core of the fundamental sphere packings. It is found that the space contains symmetry, and the structure is tetrahedral with four identical surfaces from three variables. Increased boundaries show four cube comers that make the first fragment of periodicity. Algebra offers models through its roots for natural solids, and supports the structure building principles and the models for planar defects as developed for crystals. The symmetry of the rhombic dodecahedron, with its compressed octahedron and which is the beginning of body centered packing of bodies is elaborated. It is found that saddles can be repeated using a circular function, and beautiful surfaces are obtained, topologically the same as the minimal surfaces called tower surfaces.