Fractal Analysis of Disordered Conductor–Insulator Composites with Different Conductor Backbone Structures near Percolation Threshold

Abstract

To obtain a different backbone structure of conductors in disordered insulator/conductor composites, Cu 2 O/Cu composites were prepared using two methods, named the direct hot-pressing (DHP) method and the in-situ reduction hotpressing (IRHP) method via hot-pressing Cu 2 O powders and branch-like or spherical Cu powders and hot-pressing Cu 2 O powders and carbon black (CB) nanoparticles to reduce Cu 2 O into Cu, respectively. The dc electrical conductivity tests show that the percolation threshold of the prepared Cu 2 O/Cu composites varies with various Cu structures. The lowest and the highest percolation thresholds were obtained in those with branch-like Cu and with CB nanoparticle-reduced Cu, respectively. The percolation structures of Cu were analyzed by fractal theory, and the binarized optical microscope images were used for the box-counting fractal dimension calculation. Near the percolation threshold, the fractal dimension increases with an increasing volume fraction of Cu in all prepared Cu 2 O/Cu composites. The backbone structure of the conductors was first characterized by the fractal dimension increasing rate with an increasing volume fraction of Cu. For the Cu 2 O/Cu composites prepared with the IRHP method, near the percolation threshold, the fractal dimension increases little with an increasing volume content of Cu; for those prepared with spherical Cu, the rate of increase is relatively higher, and for those prepared with branch-like Cu, the fractal dimension increase rate is the highest.

Carbon Nanofibers and Their Composites: A Review of Synthesizing, Properties and Applications

“…Fractals, as another important tool, have been widely applied to quantitatively describe the composites of randomly distributed fillers with complex and irregular geometries embedded in a matrix [ 47 ]. The fractal characterization of the fillers can be divided into regular and irregular types.…”

“…It is well known that the electrical transport properties obey the power law near the percolation threshold [ 45 ], and the geometrical structure of the fillers shows self-similarity in a scale range and is able to be characterized by fractal dimensions [ 47 ]. Numerical works demonstrated that the fractal dimensions of the infinite clusters are D ≈ 1.9 and 2.5, and the fractal dimension of the shortest path in the infinite clusters are D ≈ 1.22 and 1.43, corresponding to 2D and 3D dimensional lattices, respectively [ 62 ].…”

Dynamics of a Polymer Network Modeled by a Fractal Cactus

“…Fractals, structures with self-repeating patterns at any length scale and a non-integer dimension, are pervasive in nature and emerge in a wide variety of research areas. In physics and chemistry, the fractals are used for describing the dynamics of different polymer networks [39], porous systems [40], stretchable electronics [41], energy storage [42], disordered systems [43], growth phenomena [44], chemical reactions controlled by diffusion [45], and energy transfer [28]. The recourse to the principles of fractal geometry has enabled revealing that most biological elements, either at cellular, tissue or organic level, have self-similar structures within a defined scaling domain which can be characterized by means of the fractal dimension.…”

Dynamics of a Complex Multilayer Polymer Network: Mechanical Relaxation and Energy Transfer

“…Fractal properties have been reported for the formation of protein fibers [ 29 ] and for the organization of DNA into hierarchical structures [ 30 ]. Fractal models are widely used for describing disordered systems [ 31 ], growth phenomena [ 32 ], and chemical reactions controlled by diffusion [ 33 ]. Fractal constructs have led to the development of materials with demonstrated potentials as molecular batteries, switches, and optical display devices [ 34 , 35 , 36 ].…”