Calculations of the percolation thresholds of a three-dimensional (icosahedral) Penrose tiling by the cubic approximant method

Theoretical prediction of percolation thresholds universally applicable for various composites remains a major theoretical challenge. In the work done by Xu (2011, “Ellipsoidal Bounds and Percolation Thresholds of Transport Properties of Composites,” Acta Mech., 223, pp. 765–774), a variational method is developed to predict optimal percolation thresholds for transport properties of three dimensional composites subjected to full dispersion of fillers. In this paper, simplified formulae are provided for engineering applications of 3D composites. New formulae are derived for optimal percolation thresholds of 2D composites, i.e., laminates and thin films, and for composites containing a combination of fillers with different aspect ratios. The effects of dimensionality and waviness are especially discussed.

Download Full-text

Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field

Chinese Physics Letters ◽

10.1088/0256-307x/10/8/001 ◽

1993 ◽

Vol 10 (8) ◽

pp. 449-452 ◽

Cited By ~ 1

Author(s):

Jirong Sun

Keyword(s):

Strain Field ◽

Three Dimensional ◽

Penrose Tiling ◽

Structure Transition

Download Full-text

Kinematical electron diffraction simulations of quasicrystalline structures

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100144176 ◽

1986 ◽

Vol 44 ◽

pp. 540-541 ◽

Cited By ~ 1

Author(s):

K.M. Knowles ◽

W.M. Stobbs

Keyword(s):

Electron Diffraction ◽

Image Interpretation ◽

Three Dimensional ◽

Local Environment ◽

Icosahedral Symmetry ◽

Penrose Tiling ◽

Initial Report ◽

Occupancy Problem ◽

Diffraction Patterns ◽

Open Question

Since the initial report by Shechtman et al of a phase in a rapidly cooled Al-Mn alloy whose electron diffraction patterns exhibit icosahedral symmetry, a number of related alloys have now been discovered which exhibit this symmetry. There is now no doubt that such phases can be readily distinguished from multiply twinned particles. One interpretation of the structure of these phases is that they arise by a suitable atomic decoration of a three dimensional Penrose tiling (3DPT), in which space is filled aperiodically by prolate and oblate rhombohedra with equal sides and angles of ± acos. The diffraction patterns from the tilings themselves are known to exhibit the required symmetries. However, the details of the ‘correct’ atomic decoration remains an open question, although a variety of approaches have been tried. Solving this occupancy problem is of particular interest in HREM image interpretation, since if the appropriate decoration turns out to depend on the local environment, it will be desirable to assess the relative importance of dynamical and projection effects.

Download Full-text

The diffraction pattern of projected structures

Acta Crystallographica Section A Foundations of Crystallography ◽

10.1107/s0108767386099932 ◽

1986 ◽

Vol 42 (1) ◽

pp. 36-43 ◽

Cited By ~ 363

Author(s):

V. Elser

Keyword(s):

Diffraction Pattern ◽

Three Dimensional ◽

Delta Function ◽

Penrose Tiling ◽

Dense Set

A method for calculating the properties of structures obtained by projection is developed and applied to a three-dimensional generalization of the Penrose tiling. The diffraction pattern is shown in general to consist of a dense set of delta-function peaks. For the Penrose model the pattern in addition has the symmetry of the icosahedron.

Download Full-text